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# Chemistry Mole Packet Answer Key
375 mL of 0. 6) C8H18 C 4 H 9 7) WO2 WO 2 8) C2H6O2 CH 3 O 9) X39Y13 X 3 Y 6) A compound with an empirical formula of C. ) Unit 1 : Matter and Energy (Calorimetry) (Ch. Study Guide Chapter 2: An Introduction to Chemistry. Complete the review sheet. 20 moles of iron. For example: Common salt dissolves in water of its own. Homework: Finish notes packet. 6 mol of oxygen react? C02 c. Quiz on dot, dotlewis dot 2. Are you looking for Mole Ratios Pogil Packet Answer Key PDF Kindle to read?Mole Ratios Pogil Packet Answer Key PDF Download is highly recommended for you and Be the first to have this book!! I think the Mole Ratios Pogil Packet Answer Key ePub was fun to read and very educational. This puzzle is a great review of gas laws unit conversions and from Gas Laws Worksheet Answer Key, source: pinterest. EDGENUITY E2020 CHEMISTRY B ANSWER KEY PDF JEE Main Answer Key 2020 - National Testing Agency has released the final. Unit 5 Significant Figures Homework KEY Significant Figures W. Displaying top 8 worksheets found for - Chemistry Mole. The volume of a metal atom is 7. Mole is not a unit in the question so it is a two step problem. Whoops! There was a problem previewing Build an Atom PhET Simulation Answer Key. \ O x N A a-mec ryZ( 5. 02 x 1023 atoms of carbon. \rconvert grams to moles using molar mass\rconvert moles to volume using 22. Comments (-1) Packet #14 - Nuclear Chemistry Comments (-1) Announcements. The mole is a unit of measurement that chemists use to comprehend and count the millions and millions of tiny particles that we work with. Booklet Chapter 2. Chemical Bonding 4. net (600x819) Show images. Then, compare the model to real molecules!. In cathode-ray tubes, the cathode ray is emitted from the negative electrode, which is called the cathode. ANSWER: Example converting moles to formula units. 234 g SO 2 (1 mol SO2 64. pdf: File Size: 307 kb: File Type. Your group must reach consensus. Solutions Unit Ideal Gas Law Packet Handout Only 1-10 Review Quiz 7_mole_conversions_11-20_answer_key. If you don’t have technology and got this packet from the school, read on… Part 1: Notes 1. What volume does 22g of CO. 2 The Mole 3. Chemistry: Matter and Change Study Guide Solution 2C2H2(g) + 5O2(g) → 4CO2(g) + 2H2O(g) 20. 1 mole = 6. 5g N 1 mole N = 2. 2 inches, which contain two significant figures. How many kJ of heat are needed to completely melt 17. Begin Chapter 7: Stoichiometry by defining what it is and reviewing mole ratio 4. 00 g NaOH 8) 5. Mole-Particle Practice Worksheet 1 mdle of Partiles = 6. This is 1 mole of sulfur. Enthalpy & Mole Conversion WS (KEY) Unit 7 Review Sheet Unit 6: The Art in Chemistry (Bonding & Properties of Materials) 1st Semester Review Art in Chemistry Project Instructions - due Mon 2/10 Chemical Bonds Lab Reading/questions that accompany the lab If absent for Chemical Bonds Lab, here is the alternate assignment. Some of the worksheets for this concept are Mole calculation work, Work mole problems name, Work molemole problems name, Mole calculation work, Mole calculation work, Concentration work w 328, Mole work, Mole conversions work 1. How many particles are there in 1. Exercise Key 2. 657 g H2SO4 9) 32 g KNO3. 5 kg water Morality = 1. Calculate the amount of energy released when 1 mole of 2H fuses with 1 mole of 3H to make one mole of 4He (There is one more product. Unit 6 Packet and Key. #7 Mole Stations - Corrections Below are extra practice problems - we only did the above in class. The molar mass (gram formula mass) of a substance equals one mole of that substance. ) How To Perform Mole Calculations The Mole The Mole Overview (PPT. What 3 things (values) is a mole of a chemical equal to? There are 3 answers that you should know. 3 Electron. 0 g/rnol NH3 b. Mole-Particle Practice Worksheet 1 mdle of Partiles = 6. How many molecules of C3H70H will react with 4. Unit 5 Gas Laws Packet: File Size: Review Packet Unit 5 Gas Law Answer Key: File Size: 4060 kb: File Type: pdf. 02 x 1023 atoms of helium. Mass is the quantity of matter present; weight is a measure of the pull of gravity on matter and is measured in pounds or newtons. 02 grams of water is a just a bit more than a tablespoon-full. These are relationships and equations that you must know for the test. If this test packet is defective, ask for another one. 4 moles NaCl (aq) 9. Determine to one decimal place the molar mass of each substance and express each mass in grams per mole. 5 moles of oxygen (O 2) 7. 5 g C 6 H 12 O 6; Calculate the molarity of each of the following solutions:. (2) three moles of H2 reacts with one mole of N2. They are: 1 mole = molar mass(g) can be written as 23 1 mole = 6. 3 moles of Dinitrogen Pentoxide? 29. Unit 5 Significant Figures Homework KEY Significant Figures W. A mole is a quantity of matter that we use for conversion purposes. 8: Moles, Molecules, and Grams Worksheet. Unit 7 Pretest (partial key HERE) **full pretest key HERE** Lab Wednesday or Monday! Dress appropriately! We will take the Mole Quiz on the 8 block day on Friday! Willamette Promise Students: WP Exam 1 Practice WS (deadline for test is 3/13) key HERE **memorize metric prefixes!** Deadline for WP Exam 5: Friday 4/3. \rconvert grams to moles using molar mass\rconvert moles to volume using 22. If you have finished pages of the packet, show them to me. CHEMISTRY B- MOLES PACKET NAME: _____ HR: _____ PAGE 4 CHEMISTRY WORKSHEET # 2: THE MOLE AS A UNIT OF MASS Now that you know how to find the mass of one mole of a substance (molar mass) you can easily find the mass of several moles or the mass of a fraction of a mole using the factor-label technique. 5g N 1 mole N = 2. Base your answer to the following question on the information below Nitrogen gas, hydrogen gas, and ammonia gas are in equilibrium in a closed container at constant temperature and pressure. She includes examples and tips for effective use — and encourages teachers to incorporate these tools into their own lessons!. The volume of a metal atom is 7. 4) Solving: 1 ⁄ 5 = 1. 4 mol H 2 we need 4. Created Date: 1/22/2015 9:09:34 AM. 15 moles English muffins, 10 mole eggs, 10 mole slices of cheese and 30 moles slices of bacon _____ (continued on next page) Now, let’s connect this idea to Chemistry! Using the idea of a limiting reactant, answer the following exercises: 1. Chem1 General Chemistry Reference Text 3 Introduction to acid-base chemistry † 3 Neutralization Just as an acid is a substance that liberates hydrogen ions into solution, a base yields hydroxide ions. The coefficients in a balanced chemical equation can be used to determine mole ratios in the reaction. PROBLEMS Write the answer on the line to the left. Check out the preview for a complete view of the download. CHEMISTRY B- MOLES PACKET NAME: _____ HR: _____ PAGE 4 CHEMISTRY WORKSHEET # 2: THE MOLE AS A UNIT OF MASS Now that you know how to find the mass of one mole of a substance (molar mass) you can easily find the mass of several moles or the mass of a fraction of a mole using the factor-label technique. answer key to chemistry b moles packet bing just pdf. Always use units and box your final answer. 2% carbon? (Answer: C4H9) 39. 1 Atoms and molecules are too small to count. This puzzle is a great review of gas laws unit conversions and from Gas Laws Worksheet Answer Key, source: pinterest. The mole, also known as Avogadro’s number, is equal to 6. Connect this to the analysis of a potential source of fuel. Chemistry Lab Manual® Tenafly High School ©September 2010. Unit 5 Gas Laws Packet: File Size: Review Packet Unit 5 Gas Law Answer Key: File Size: 4060 kb: File Type: pdf. this can be written as the ratio (1 N2)/(2 NH3) Again, we can write the inverse of this: (4) two moles of NH3 are produced from one mole of N2. NCERT Solutions for Class 11 Chemistry Chapter 6 Short Answer Type Questions. Founded in 1900, the College Board was created to expand access to higher education. 07 ÷ 2 × 22. What is the mole ratio of oxygen to carbon dioxide? q Oa ID COA b. ) Hints: Always begin by writing out the formula for any compound in the problenz. 400 mole of #23 weigh? 63. 0g CH2 2 2 2 2 2 1 mol CH 26. Read and Download Ebook Nuclear Chemistry Packet Answer Key PDF at Public Ebook Library NUCLEAR CHEMISTRY PACKET ANSWER KEY PDF DOWNLOAD: NUCLEAR CHEMISTRY PACKET ANSWER KEY PDF Give us 5 minutes and we will show you the best book to read today. pdf : UNIT 10: Naming Organic Compounds U10 HW KEYS. pdf: File Size: Download File. 2 moles of hydrogen and 1 mole of Chapter 12 Stoichiometry Packet Answers Read PDF Stoichiometry Packet Answers Chapter 12 We'll practice limiting reactant and excess reactant by working through a problem. Unit 7 Pretest (partial key HERE) **full pretest key HERE** Lab Wednesday or Monday! Dress appropriately! We will take the Mole Quiz on the 8 block day on Friday! Willamette Promise Students: WP Exam 1 Practice WS (deadline for test is 3/13) key HERE **memorize metric prefixes!** Deadline for WP Exam 5: Friday 4/3. net (600x819) Show images. Video 11: Reaction Stoichiometry. The Molar Mass (aka gram formula mass or molecular mass) is the mass of one mole of a substance. Lab Quiz 2. Unit 4 bonding work packet answer key; Chemistry unit 5 the mole key; Unit 5 the mole and stoichiometry answer key; Recent Homework Help Questions from CLEP Test. A gas occupies a volume of 35. If this test packet is defective, ask for another one. Moles Lab Activities - Virginia Department of Education. Chemistry B January 13: Solubility of salt lab Work on notes packet (slides 10-25), and pass back tests molarity_and_stoichiometry_homework_answer_key. Chemistry Practice Test: Ch. Chapter 2: An Introduction to Chemistry. Element Names and Symbols. 4 x 1024 molecules of NH3?96 grams 5) How much does 4. Are you looking for Mole Ratios Pogil Packet Answer Key PDF Kindle to read?Mole Ratios Pogil Packet Answer Key PDF Download is highly recommended for you and Be the first to have this book!! I think the Mole Ratios Pogil Packet Answer Key ePub was fun to read and very educational. The Simple Machines Unit is now over 50+ pages. 2 Mole-Mass and Mole-Volume Relationships. 02 x 1023 Pañticles (Particle is the gene?ic word that we use in chemistry for: molecule,fonnula unit, ion, atom, etc. Mole Concept Answer Key. Ideal Gas Law Constant R 0. ) The required masses are: 2H: 2. This is it, the Nuclear Chemistry Packet Answer Key that will be your best choice for better reading. Through years of experimentation, it has been established that a mole of anything contains. pdf : UNIT 9 Chemical Names & Formulas U9 HW Keys. Atomic Parts Worksheet Atoms and Ions Worksheet Radioactivity and Half Lives 10/4 Periodic Table Vocabulary 10/5 Periodic Table Spreadsheet and Questions, Periodic Table worksheets - due Monday Individual Atomic Structure SS Mole Problem Packet due - check your answers with the key. o The entire packet is due on 9/4 (A) and 9/5 (B). 1 Chemical Configuration and Isotopes 3. Chapter 1: An Introduction to Chemistry. pdf Kinetics Exam Answer Key. What is the volume of the combined gases at STP. 1 grams 2) H 3PO 4 3 + 31 + 64 = 98. 75 moles of water will be produced. Rententber to give your answer to the correct number of significant digits. 3_answer_key. How much does 0. o Complete each worksheet in this packet. A sample of nitrogen gas. Calculate the mole fraction, molarity and molality of NH3 if it is in a solution composed of 30. Use this download to review or assess your students' understanding of mole conversions and calculations. Write and balance the chemical equation. Unit 7 Moles & Stoichiometry Notes Packet Unit 7 Moles & Stoichiometry Practice Packet Practice Test: Moles & Stoichiometry (Practice Test: (Answer Key) Moles & Stoichiometry Stoichiometry Review KEY Stoichiometry Review. this can be written as the ratio (1 N2)/(2 NH3) Again, we can write the inverse of this: (4) two moles of NH3 are produced from one mole of N2. Quiz: Stoichiometry/% Yield/Limiting Reac. 2 g O2 are reacted. 5 Limiting Reactant Problems 3. 5 moles of oxygen (O 2) 7. E: Homework Chapter 8 Answer Key Last updated; Save as PDF Page ID 202175; No headers. Unit 8 Short Answer Packet Key File. Activity 8. pdf: File Size: 42 kb: File Type: pdf. around the most probable is narrowing dramatically. Question 1. 6) C8H18 C 4 H 9 7) WO2 WO 2 8) C2H6O2 CH 3 O 9) X39Y13 X 3 Y 6) A compound with an empirical formula of C. 2 The Mole 3. Do not depend on answer keys to do your homework. C3H70H + + a. 3 grams of H 2 S could be produced. On this page you can read or download chemistry b moles packet key in PDF format. O 2 / H 2O d. Rententber to give your answer to the correct number of significant digits. The mole is a quantity like a dozen (12) or a gross (144). 500 mL of 0. Day 1 worksheet - phase diagrams and pressure unit conversions; answer key is on pages 3-4 Day 2 worksheet - Charles's law and Graham's law of diffusion; answer key is on pages 3-4 Gas law lab (pressure, volume and temperature) - if you were absent, use this data on the last page. ANSWER: Example converting moles to formula units. Determine the hybridization of the central atom, the VSEPR formula, the geometric shape, bond angles, and the polarity of the molecule. Mole Ratio Worksheets - Lesson Worksheets Mole Ratios Pogil Packet Answer Key PDF Kindle. Answers is the place to go to get the answers you need and to ask the questions you want. How many grams are in 2. exact -counting objects (12 apples), definitions (100 cm in a meter) inexact -any measured length, weight, temperature, etc. 3/3 - 3/6 Mole Packet Molecular vs Empirical Formulas Notes Pg 21-23 PPT 1-29 Molecular vs Empirical Formula. What happened. How many moles of carbon dioxide are produced when 4. How many moles are contained in 1. 1 Ionic and Covalent Bonding 4. Work on Writing Formulas Packet p44-48, due tomorrow. W:: This lesson is an introduction to the concept of the mole and why chemists use it. Title: Relative Mass and the Mole answer key Created Date: 20171005134609Z. CHEMISTRY B- MOLES PACKET NAME: _____ HR: _____ PAGE 4 CHEMISTRY WORKSHEET # 2: THE MOLE AS A UNIT OF MASS Now that you know how to find the mass of one mole of a substance (molar mass) you can easily find the mass of several moles or the mass of a fraction of a mole using the factor-label technique. In this case, measurements made by that ruler would have only one significant figure (1 inch or 6 inches, as opposed to 1. Step 3 Determine the number of moles of the unknown substance, using mole-to-mole conversion. pdf Kinetics Exam. Mole Conversions Practice Gap-fill exercise. Unit 6 Packet KEY. Mole Conversions/Chemical Quantities This is an excellent site for all things chemistry: Answer key for Colligative Property Chemquest. chemistry worksheet # 6 mixed mole problems (grams, molecules, and liters) You now know three things a mole can be: a molar mass, 6. this can be written as the ratio (3 H2)/(1 N2) (3) one mole of N2 produces two moles of NH3. 1 Atoms and molecules are too small to count. 3 g of O, given that the water is at its melting point? The heat of fusion for water is 6. Unit 11 Packet - Page 8 of 14 GAS LAW PROBLEMS Work the following problems and identify the gas law used; be sure your answer includes units! 1. A compound with an empirical formula of CFBrO and a molar mass of 254. Nuclear Chemistry Note Taking Sheet; Nuclear Fission/Fusion Webquest. While there is certainly a subset of the American prison. What is the mole ratio of oxygen to carbon dioxide? q Oa ID COA b. Just preview or download the desired file. 1-Edpuzzle videos about Mole Concept Answers for Winter Break Study Guide Packets for Benchmark Exam : HERE1 and HERE2 Answers for Mole Concept worksheet HERE3 Week-16: 12/16/2016: Winter Break Study Guide Packets for Benchmark Exam: Packet:1 HERE and Packet:2 Here Unit-5 NOTES HERE Classroom Activities Answers: Here:2 and Here:3 Week-15: 12/9. Empirical and Molecular Formula Worksheet ANSWER KEY. answer key to chemistry b moles packet bing just pdf. 2 POGIL™ Activities for High School Chemistry 1. Many students can not wait to see something "blow up" in the lab, but this is actually what we try to avoid in a chemistry classroom! Hopefully by the end of this unit they will have a greater appreciation for other types of reactions like this copper coil soaking in a solution of silver nitrate. This is 1 mole of sulfur. We can use this very helpful unit to convert not-very-helpful units of measurements into ones that we can use in lab on a daily basis. 022x1023particles((atoms,(molecules)( Microsoft Word - Ch 9 review Answer Key. Percent Composition Review Book HW Answers AND Mole Conversions Packet and Answer Key Assigned as HW on 2/28/18 Stoichiometry WS 1 and 2 Answer Key. 4 L\r standard temperature and pressure - clue you will use molar volume. Mole to Mole Wks. 2% carbon? (Answer: C4H9) 39. 4 L O2 1 mol O2) = 0. exact –counting objects (12 apples), definitions (100 cm in a meter) inexact –any measured length, weight, temperature, etc. Unit 6 Packet. Chemistry Interactive Review Activities. Rententber to give your answer to the correct number of significant digits. 4 moles of sulfur? 77. com/balance. That is why Peterson’s is everywhere education happens. L, mL = liter(s), milliliter(s) mm Hg = millimeters of mercury g = gram(s) J, kJ = joule(s), kilojoule(s) nm = nanometer(s)V = volt(s) atm = atmosphere(s) mol = mole(s) ATOMIC STRUCTURE E = hQ. pdf : UNIT 8: Covalent Bonding U8 HW Packet Key. (2) three moles of H2 reacts with one mole of N2. Click here for answer key to review worksheet 4. Unit 6 Packet KEY. (Coefficients equal to one (1) do not need to be shown in your answers). 2 Mole-Mass and Mole-Volume Relationships. 7 moles of. 022\times {10}^{23}[/latex] of something, but it can be difficult to conceptualize such a large number. pdf UNIT 6 Organizing the elements & Trends in the periodic table U6 HW Packet Key. Find the empirical formula of a compound that contains 15. General Questions. 4 liters at STP. ) Hints: Always begin by writing out the formula for any compound in the problenz. A gas occupies a volume of 35. Thus, CaCl 2 will generate more ions per mole than 1 mole of NaCl and be a better conductor of electricity. Manipulate a conceptual model to understand how the number of particles, the number of moles, and the mass are related. O 2 / H 2O d. [Filename: regents topic review packet1. How many grams are in 2. If you don't see any interesting for you, use our search form on bottom ↓. 4 L at STP ) 1. 1490 Roth's Church Road, Spring Grove, PA 17362. Answers without such work receive no credit. Through years of experimentation, it has been established that a mole of anything contains. glencoe chemistry matter change answer key chapter 12 Media Publishing eBook, ePub, Kindle PDF View ID c53aaac86 Mar 07, 2020 By Andrew Neiderman chemistry chapter 12 assessment answers document preview fishhawk grade 4online reading math. What is the mass of exactly one mole of calcium acetate? Ca(C 2 H 3 O 2) 2 158g/mol 22. Mole to Mole Wks. The key factor here turns out to be the interaction of the ions with the solvent. 10/21: Mole Quiz and Intro to Mole Ratios 10/22: Mass Stoichiometry CLICK HERE for notes 10/23: S'more Lab and Percent Yield 10/24: Complete Mass Stoichiometry questions- key in files below (stoichiometry practice packet). Thanks to the authors of the HotPotatoes program for making this possible!. Final Practice examination answer Key 3 Grade 11 C hemistry (30s) F ˆ ˘ P˛ ˜ E! ˇ ˆ ˜ ˙ˆ Aˆ˚ ˛#K " II c the final examination will be weighted as follows modules 1 –3 15 –20% modules 4 –6 80 –85% the format of the examination will be as follows: Part a: Fill-in-the-Blanks 22 x 1 = 22 marks Part B: multiple Choice 46 x 1 = 46. Mole to Mole Practice problems - Answers: 11. 100 g of oxygen(O 2) is added to the gas in Question 16. Worksheet Template Uncategorized Mole Ratio Worksheet Answers from Mole Ratio Worksheet, source: maveno. Both have answers. Begin Chapter 7: Stoichiometry by defining what it is and reviewing mole ratio 4. These things can be atoms, or molecules, or eggs; however, in chemistry, we usually use the mole to refer to the amounts of atoms or molecules. Thus, CaCl 2 will generate more ions per mole than 1 mole of NaCl and be a better conductor of electricity. pdf: File Size: Download File. 7th edition (sh) Lab # CHEMISTRY LAB - ACTIVITY TITLES Page 1 Lab Safety 5 2 Matter Classification 7 3 Classifying Physical and Chemical Changes 9 4 Chemical Properties of Four Liquids 11 5 Density of Pennies 13 6 Percent Cu in Pennies 15 7 Law of Definite Composition 17 8 Particle Size Probability 19 9 Emission Spectroscopy 21 10. Wednesday, Discussed Packet from Monday, Atoms - Clash of the Titans Questions; Extra Practice Density Problems Answer Key. 0 sections 8. Other Results for Chapter 10 Chemical Quantities Test Answer Key: Chemistry I H: Chapter 10 Chemical Quantities- Chapter Chapter 10: Chemical Quantities- Chapter Test B (pages 256-259) by Pearson Education Learn with flashcards, games, and more — for free. From above we can see that if we have 12. 1 Chemical Configuration and Isotopes 3. Mole-Particle Practice Worksheet 1 mdle of Partiles = 6. H:: The lesson begins with a warm-up to activate prior knowledge on atomic mass. A mole is defined as the number of carbon atoms in exactly 12 g of pure carbon-12. Percent Yield - The Chemistry Solution: Wed. 02 x 1023 atoms of sodium. 2 moles of oxygen and 1 mole of hydrogen in the products : b. Expressed in scientific notation, a mole is 6. It will be collected as a formative assignment on test day. Unlike traditional textbooks, this Rapid Learning series will introduce you the chemistry via visual learning and smart teaching. Chemistry-1 Practicing the Mole - - Odd Problem Answer Key Page 1 Practicing the Mole - - Even Problems Answer Key Calculate the mass in grams of each of the following: 2. Study Guide for AP Chemistry. 2? g : 32 5. 02 x 1023 atoms of carbon. Today, the membership association is. org Moles and Molar Mass The mole represents 6. Chapter 1: An Introduction to Chemistry. Friday: Quiz. 1 Classification 5. 8: Moles, Molecules, and Grams Worksheet. Activity 8. The equation below represents this equilibrium. Nuclear Chemistry. O 2 / H 2O d. Unlike traditional textbooks, this Rapid Learning series will introduce you the chemistry via visual learning and smart teaching. \ O x N A a-mec ryZ( 5. with Answers: 12. Answer any questions that arise. Limiting reagent stoichiometry Get 5 of 7 questions to level up! Molecular composition. Mole Ratios Pogil Packet Answer Key PDF Download is highly recommended for you and Be the first to have this book!! I think the Mole Ratios Pogil Packet Answer Key ePub was fun to read and very Mole Ratios Pogil Packet Answer Key PDF Kindle HS Chemistry POGIL Activity Topic: show how you calculated your answer. Exercise Key 2. Now CAREFULLY place the alka-seltzer bag inside of the water bag. Look at Figure 16. If you wanted to know how. Analysis GramFormulaMass Moles of Elements Mole Guide Mole-Mole Pract Probs Some Mole Problems More Mole Problems Mol-Avagadro Mol-Mass Mol-Volume Mixed Mole Percent Comp Molarity Ch 10 Pract. has a "mass incarceration" problem; if it does, then mass decarceration is the obvious solution. 3 Mole to Mole Problems 3. Percent Yield - The Chemistry Solution: Wed. 45 x 1023 molecules of CH4?0. • For chemists, stoichiometry is a form of bookkeeping. 2 moles of H 2 is the correct answer. 64 moles C02 28 g C02 44. Answer: A process which can take place of its own or initiate under some condition. Wednesday, Discussed Packet from Monday, Atoms - Clash of the Titans Questions; Extra Practice Density Problems Answer Key. Chemistry Home Honors Chemistry hw_20. Ch 10 Notes & Examples from Class. Structure of Matter. Chemistry in a Bag Report (Intro and Hypothesis) DUE FRIDAY 9/20/13*** Study for Mole Conversion Test Electron Configuration Worksheet Vocab Quiz 2 corrections*** Friday September 20, 2013 Cl-Ev-R Report ( Nuclear Chemistry) - Due MONDAY 9/23/2013*** Monday September 23, 2013 Wanted! Element Project Due 9/30 Empirical Formula Worksheet. BE SURE TO BALANCE EACH EQUATION BEFORE SOLVING ANY PROBLEMS. Review all. If you don’t have technology and got this packet from the school, read on… Part 1: Notes 1. You must show all work! Mole. 02 x 1023 atoms of gold. Kindly say, the Mole Ratio Lab Answers is universally compatible with any devices to read Guided Reading And Study Workbook Chapter 10 Answer Key, Answer Key Pathways 2 Reading And Writing, western democracy guided reading key, Reading Images The Grammar Of Visual Design Gunther Kress, Chapter 18 Section 2 Guided Reading The. 8: Moles, Molecules, and Grams Worksheet. E: Homework Chapter 8 Answer Key Last updated; Save as PDF Page ID 202175; No headers. AP Chemistry: Electrochemistry Multiple Choice Answers 14. 0500 mole of N02 (g), 0. 02/27/20: 02/25/20: Complete the Stoichiometry - Mole Ratio Worksheet on page 1 of the packet. Honors Chemistry Extra Stoichiometry Problems 1. 4 mol H 2 we need 4. o Show all work/ justify your answers. Solutions Unit Ideal Gas Law Packet Handout Only 1-10 Review Quiz 7_mole_conversions_11-20_answer_key. Questions 1-4 show pulses A and B at time = 0 as they head toward each. 50 g of nitrogen (N 2) has a volume of ___ liters at STP. No Warm Up Question. Video 11: Reaction Stoichiometry. exact –counting objects (12 apples), definitions (100 cm in a meter) inexact –any measured length, weight, temperature, etc. In the front of the room, there is a bottle that contains a 32. 2 g O2 are reacted. photosynthesis lab answer key. Experiments established that number to be 6. 016029 amu; 4He: 4. >>>CLICK HERE<<<. pdf: File Size: 1002. Answers is the place to go to get the answers you need and to ask the questions you want. However, we have learned about seven elements that exist as diatomic molecules—H 2, N 2, O 2, F 2, Cl 2, Br 2, and I 2. 3 Electron. 3_answer_key. Chemistry Lesson: The Metric System & Conversions The Metric System The metric system was created by the French in the late 1700’s in response to the English system which, quite frankly, is fairly confusing in that it uses numerous different units and has no uniformity among conversions. use your knowledge of gas laws and stoichiometry to design Gas Laws. #7 Mole Stations - Corrections Below are extra practice problems - we only did the above in class. S-2 The story takes place. Answer this question and win exciting prizes. A mole of helium contains 6. 8: Moles, Molecules, and Grams Worksheet. Instructions Before viewing an episode, download and print the note-taking guides, worksheets, and lab data sheets for that episode, keeping the printed sheets in order by page number. Words may be used more than once or not used at all. 500 mL of 0. CHEMISTRY B- MOLES PACKET NAME: _____ HR: _____ PAGE 8 CHEMISTRY WORKSHEET # 5 MOLE PROBLEMS—MOLAR VOLUME OF A GAS We have learned that a mole is a mass of material and number of particles. 0 sections 8. SHOW the math below. How many particles are there in 1. You might not require more epoch to spend to go to the books instigation as skillfully as search for them. Choose the one alternative that best completes the statement or answers the question. 6 g NH3 in 81. 1 Expert Answer(s) - 228989 - a food packet is dropped from a helicopter rising up with vel. Element Properties. One mole of glycine, C 2 H 5 O 2 N, contains 2 moles of carbon, 5 moles of hydrogen, 2 moles of oxygen, and 1 mole of nitrogen: The provided mass of glycine (~28 g) is a bit more than one-third the molar mass (~75 g/mol), so the computed result is expected to be a bit greater than one-third of a mole (~0. 8: Moles, Molecules, and Grams Worksheet. ANSWER = NO2. Given the following equation: N 2(g) 3H 2(g) → 2NH 3(g) 28. B)the energy required to convert a mole of ionic solid into its constituent ions in the gas phase C)the sum of ionization energies of the components in an ionic solid D)the energy given off when gaseous ions combine to form one mole of an ionic solid E)the energy required to produce one mole of an ionic compound from its constituent elements in. 578 KB (Last Modified on July 8, 2016) Comments (-1). Complete the practice problems on page 4-5. 125 M K 2 SO 4 21. Chemistry Practice Test: Ch. KEY Molarity: • a _____ description of solution concentration. Limiting reagent stoichiometry Get 5 of 7 questions to level up! Molecular composition. Words may be used more than once or not used at all. Stoichiometry Ch 12 Packet Answer Key Stoichiometry Ch 12 Packet Answer Chapter 12 Stoichiometry Packet Answers Stoichiometry Ch 12 Packet Answer Key Stoichiometry Ch 12 Packet Answer misterchemistrycom Chapter 12 Stoichiometry In the reaction represented by the equation 2Na 2H20 —y 2NaOH + H2, how many grams of hydrogen are produced if. 9 g/mol : 9. 0821 L*atm to solve the following problems: K*mol If pressure is needed in kPa then convert by multiplying by 101. mole problems chemistry if8766 answers PDF may not make exciting reading, but mixed mole problems chemistry if8766 answers is packed with valuable instructions, information and warnings. EDGENUITY E2020 CHEMISTRY B ANSWER KEY PDF JEE Main Answer Key 2020 - National Testing Agency has released the final. Home Honors Chemistry In Class Study Guide- Answer Key. The Summer Assignment is included in this packet. Chemistry: Matter and Change Study Guide Solution 2C2H2(g) + 5O2(g) → 4CO2(g) + 2H2O(g) 20. FREE OpenStax Chemistry Textbook. Mole Ratios Pogil Packet Answer Key PDF Kindle. 2 POGIL™ Activities for High School Chemistry 1. Step 4 Determine the mass of the unknown substance, using mole-to-mass conversion. 3 L of a gas weighs 2 g. [Filename: regents topic review packet1. Define a mole. pdf : UNIT 11: Chemical Quantities - The MOLE U11. Review Mole Ratio by going over first set of problems WS 7. Click here for answer key to review worksheet 4. 02 mol H2O 3) 0. Potassium chlorate decomposes into potassium chloride and oxygen gas. 375 mL of 0. Lecture/ Presentation Click on the image above to access the instructional PowerPoint that is covered inc lass. Week 11-13. TanyaElias. Most likely you have knowledge that, people have see numerous time for their favorite books with this Mole Ratios Packet Answer Key, but stop up in harmful downloads. 5 moles of Nitrogen Gas? 28. (Answer: A12S3012 or 40. Chapter Two. Honors Chemistry: General Chemistry: Introduction to Chemistry (Safety Rules) (Equip. Answer any questions that arise. 6 mol of oxygen react? C02 c. Digging the Mole Concept in Chemistry. TanyaElias. 8 g K 2 SO 4 b. A sample of gas has a volume of 215 cm3 at 23. What volume does 22g of CO. STOICHIOMETRY MOLE - MOLE. You must show work in the spaces provided that leads to your answers to problems 2 and 3. How many grams are in 3 moles of Water? 27. Frei's Chemistry Class: Welcome, Knights! Mole Ratios. 7 moles of. Murdoch Page 9 of 61 Website upload Key The hydrated (aqueous) ions are kept separated by the motion of the water molecules that are attracted to them. Burn the Biofuel. File Type PDF Molarity Packet Answers Pogil Pogil Packet Answer Key PDF Kindle substance in the reaction is called the mole ratio. Solute: 190 g CUS04 1 mole = I. Moles Lab Activities - Virginia Department of Education. 0 sections 8. Do Mr Guch's "Moles Molecules, and Grams" and "Mole Calculation" worksheets. What volume will the gas occupy at STP? 4. 43 g of a molecular compound with a gam. W:: This lesson is an introduction to the concept of the mole and why chemists use it. 1) Formation of solutions where the process is endothermic can be spontaneous provided that _____. welcome back This video contains 00:23 Electronic Configuration 01:29 Number Of Orbitals 02:12 Subshells 04:19 Finding Blocks 06:22 Periodic Table 07:06 Finding Periods 08:16 S Block 09:23 Valency. 1 answer First, we need to know a little bit about water. Home Science Math History Literature Technology Health Law Business All Topics Random. 4 moles of sulfur? 77. ADVANCED PLACEMENT CHEMISTRY EQUATIONS AND CONSTANTS Throughout the test the following symbols have th e definitions specified unless otherwise noted. 02 X 10 MOLECULES (for compounds) Atomic Weight (for elements) or Gram Formula/Molecular Weight (for compounds) MOLES 22. Mole Ratios Packet Answer Key Mole Ratios Packet Answer Key Thank you very much for downloading Mole Ratios Packet Answer Key. Mass is the quantity of matter present; weight is a measure of the pull of gravity on matter and is measured in pounds or newtons. (Answer: A12S3012 or 40. Author: Trevor Created Date: 5/17/2015 6:33:38 PM. Relationships Answer Key Unit 7 Mole Relationships Answer Key This is likewise one of the factors by obtaining the soft documents of this unit 7 mole relationships answer key by online. CHEMISTRY B- MOLES PACKET NAME: _____ HR: _____ PAGE 8 CHEMISTRY WORKSHEET # 5 MOLE PROBLEMS—MOLAR VOLUME OF A GAS We have learned that a mole is a mass of material and number of particles. Answer Key 3 Chemistry Chapter Equations Balancing. 58 g of the substance in #23? 0. CHEMISTRY is the study of matter, its properties, how and why substances combine or separate to form other substances, and how substances interact with energy. Manipulate a conceptual model to understand how the number of particles, the number of moles, and the mass are related. Mole is the unit used to tell how many particles are in a certain amount of a substance. The Mole and Volume Worksheet (DOCX 15 KB) Weekly 6 Homework (DOC 52 KB) Weekly 7 Homework (DOC 55 KB) Mole Test - Review Packet (DOCX 18 KB) Mole Test - Review Packet - Answer Key (DOCX 27 KB) Stoichiometry- Mole-Mole Problems Worksheet - Answer Key (DOCX 16 KB) Stoichiometry - Volume-Volume Problems Worksheet - Answer Key (DOCX 18 KB) NEED. Tomorrow I will post a video going over a few of the problems! If you don't have a printer, you may hand write these. criss-cross charges for ionic compounds!!!) Use the mole ratios from the balanced equations to solve the following stoichiometry problems. free downloadchemistry b moles packet answer key. What is the mole ratio of oxygen to carbon dioxide? q Oa ID COA b. Make sure to show the dimensional analysis in your work. Final Practice examination answer Key 3 Grade 11 C hemistry (30s) F ˆ ˘ P˛ ˜ E! ˇ ˆ ˜ ˙ˆ Aˆ˚ ˛#K " II c the final examination will be weighted as follows modules 1 –3 15 –20% modules 4 –6 80 –85% the format of the examination will be as follows: Part a: Fill-in-the-Blanks 22 x 1 = 22 marks Part B: multiple Choice 46 x 1 = 46. 2 The Mole 3. Read and complete textbook pages 102-109. Chapter 13-14: States of matter and gas laws Mole % by mass HW KEY. SEMESTER REVIEW PACKET DUE. pdf : UNIT 9 Chemical Names & Formulas U9 HW Keys. an exam after instruction in which there is an answer key with the 'correct' answer. 02/26/20: 02/24/20: Complete the 3 problems that we discussed in class on the S'more Stoichiometry worksheet. ½ Life Activity with Lab Write-up Poster summary: Review Sheet Use the Review Sheet to help you Select five key aspects from this unit that they feel are most significant to the understanding of Atomic Theory. EDGENUITY E2020 CHEMISTRY B ANSWER KEY PDF JEE Main Answer Key 2020 - National Testing Agency has released the final. 2 AgNO 3 + BaCl 2! 2 AgCl + Ba(NO 3) 2 b. 022x1023particles((atoms,(molecules)( Microsoft Word - Ch 9 review Answer Key. Quiz: Stoichiometry/% Yield/Limiting Reac. To communicate the exactness of our measurements. Given the Ibilowing eouation: N(g; ± 3H ç — 2NH3gl 28. A mole of sodium contains 6. 4) Solving: 1 ⁄ 5 = 1. 717-225-4731 717-225-0736. 1 Ionic and Covalent Bonding 4. Answer the following questions: 9. A compound with an empirical formula of C2H8N and a molar mass of 46 grams per mole. Comments (-1) Packet #14 - Nuclear Chemistry Comments (-1) Announcements. 1 - Molecules & Elements 2 - Chemical Analysis 3 - The Mole 4 - Coulomb's Law 5 - Electron Configuration 6 - Periodicity. 10/21: Mole Quiz and Intro to Mole Ratios 10/22: Mass Stoichiometry CLICK HERE for notes 10/23: S'more Lab and Percent Yield 10/24: Complete Mass Stoichiometry questions- key in files below (stoichiometry practice packet). Different mole ratios occur for other polyprotic acids or bases with multiple hydroxides such as $$\ce{Ca(OH)_2}$$. The coefficients in a balanced chemical equation can be used to determine mole ratios in the reaction. Analyze the pros and cons of producing biodiesel. reactions is a subject of chemistry called stoichiometry. Chemistry Lab Manual® Tenafly High School ©September 2010. Name three factors that influence the rate at which a solute. 1) Formation of solutions where the process is endothermic can be spontaneous provided that _____. B)the energy required to convert a mole of ionic solid into its constituent ions in the gas phase C)the sum of ionization energies of the components in an ionic solid D)the energy given off when gaseous ions combine to form one mole of an ionic solid E)the energy required to produce one mole of an ionic compound from its constituent elements in. com The Mole and Avogadro’s Number Worksheet (DOCX 18 KB) The Mole and Volume Worksheet (DOCX 15 KB) Weekly 6 Homework (DOC 52 KB) Weekly 7 Homework (DOC 55 KB) Mole Test - Review Packet (DOCX 18 KB) Mole Test - Review Packet - Answer Key (DOCX 27 KB) Stoichiometry- Mole-. You might not require more times to spend to go to the ebook instigation as well as search. Are you looking for Mole Ratios Pogil Packet Answer Key PDF Kindle to read?Mole Ratios Pogil Packet Answer Key PDF Download is highly recommended for you and Be the first to have this book!! I think the Mole Ratios Pogil Packet Answer Key ePub was fun to read and very educational. 122 g C12H22O11 x (1 mol C12H22O11/ 342. study-guide2 (answer key) i have everything answered, except the symbols, it would not work on my computer. Na2HP04 12H20. Chemistry Practice Test: Ch. Unit 6 Packet and Key. pdf Kinetics Exam. 5 Limiting Reactant Problems 3. 2 Mole-Mass and Mole-Volume Relationships. 2mol H 2 2 mol NH 3 2 mol NH 3 Key Questions: 1. A mole of carbon contains 6. 02/25/20: 02/21/20. Unit 5: Moles. Questions 1-4 show pulses A and B at time = 0 as they head toward each. 2H20 and coC12. 07 ÷ 2 × 22. Chemistry Mole. 00 mol NH3 2) 9. 350 M C 6 H 12 O 6 31. 016029 amu; 4He: 4. 2 HW: --finish WS 7. Physical Science: Chemistry Gas laws unit packet. 8: Moles, Molecules, and Grams Worksheet. Use the answer key on page 5-6 to check your. Click here for answer key to review worksheet 4. Where did the 3 𝑜 𝐻2 2 𝑜 𝑁𝐻3, come from? 3. 02 grams of barium chloride are reacted in an excess of silver nitrate, how many. 326g of C6H6 dissolve in 820. C3H70H + + a. Select your preference below and click 'Start' to give it a try!. 4 L at STP ) 1. 2 Lewis Structures 4. Chemistry B January 13: Solubility of salt lab Work on notes packet (slides 10-25), and pass back tests molarity_and_stoichiometry_homework_answer_key. Amedeo Avogadro's work on gases helped define a new unit of measurement for chemistry and physics: the mole. At a mole of H and a mole of D, for example, it would be a sharp line. For any questions that require math, write out any equations. 375 mL of 0. L, mL = liter(s), milliliter(s) mm Hg = millimeters of mercury g = gram(s) J, kJ = joule(s), kilojoule(s) nm = nanometer(s)V = volt(s) atm = atmosphere(s) mol = mole(s) ATOMIC STRUCTURE E = hQ. Percent Yield - The Chemistry Solution: Wed. S-2 The story takes place. Mole is not a unit in the question so it is a two step problem. CHEMISTRY B- MOLES PACKET NAME: _____ HR: _____ PAGE 8 CHEMISTRY WORKSHEET # 5 MOLE PROBLEMS—MOLAR VOLUME OF A GAS We have learned that a mole is a mass of material and number of particles. Questions 14-17 The spontaneous reaction that occurs when the cell in the picture operates is as follows: 2Ag+ + Cd (s) à 2 Ag (s) + Cd 2+ (A) Voltage increases. Stoichiometry Notes - Mole Ratios/Mole to Mole Conv. mass-to-mole conversion. when allowed to stand or when stirred? c. AP Chemistry. SHORT ANSWER Answer the following questions in the space provided. Using Equations Balanced Chemical Equations. Click on pop-out icon or print icon to worksheet to print or download. Chapter 13-14: States of matter and gas laws Mole % by mass HW KEY. O 2 / CO 2 c. Nuclear Chemistry Note Taking Sheet; Nuclear Fission/Fusion Webquest. stoichiometry worksheet answers – streamcleanfo from Gas Laws Worksheet Answer Key, source. However, we have learned about seven elements that exist as diatomic molecules—H 2, N 2, O 2, F 2, Cl 2, Br 2, and I 2. 375 mL of 0. 2007B-2 Key : 2009-6 2009-6 Key 2018-7:. 00 moles of mercury 3. as a whole cube or in granulated form? b. Mole ratio Answer: 34. 1 - Molecules & Elements 2 - Chemical Analysis 3 - The Mole 4 - Coulomb's Law 5 - Electron Configuration 6 - Periodicity. CHEMISTRY B- MOLES PACKET NAME: _____ HR: _____ PAGE 8 CHEMISTRY WORKSHEET # 5 MOLE PROBLEMS—MOLAR VOLUME OF A GAS We have learned that a mole is a mass of material and number of particles. 1 Molecular Weight and Mass Percent 3. net (600x819) Show images. Final Practice examination answer Key 3 Grade 11 C hemistry (30s) F ˆ ˘ P˛ ˜ E! ˇ ˆ ˜ ˙ˆ Aˆ˚ ˛#K " II c the final examination will be weighted as follows modules 1 –3 15 –20% modules 4 –6 80 –85% the format of the examination will be as follows: Part a: Fill-in-the-Blanks 22 x 1 = 22 marks Part B: multiple Choice 46 x 1 = 46. Step 2: Divide smallest mole number in each element to get ratio of that element. PROBLEMS Write the answer on the line to the left. 3_answer_key. C 4H 10 / H 2O 2. Chemistry Stoichiometry Packet Answers Chemistry Stoichiometry Packet Answers Chemical Reactions and Stoichiometry Given the equation 3A + B → C + D, you react 1 mole of A with 3 moles of B True or false: B; is the limiting reactant because you have. A mole of a particular substance is equal to the number of atoms in exactly 12 g of the carbon-12 isotope. Molarity: 15. 015 M NaF 0. _____ H2 (g) + _____ O2 (g) ( _____ H2O (g) Given the equation above, determine the number of moles of water produced when 5. Nuclear Chemistry Study Guide AnswerKey. A mole of basketballs would just about fit perfectly into a ball bag the size of the earth. 6 x 10 5 J Note: It makes sense that ΔG° is negative because this reaction is thermodynamically-favorable. Mole Ratio - Displaying top 8 worksheets found for this concept. 375 mL of 0. My Favorite Technology Tools for Teaching Chemistry — Remotely and In-Person. when allowed to stand or when stirred? c. Estimate how many atoms are in the bottle. 2 HW: --finish WS 7. Collision theory, theory used to predict the rates of chemical reactions, particularly for gases. STOICHIOMETRY PROBLEMS Answer Key (continued) Empirical Formulas of Hydrates, p. com by Monday, 5/4, at 9am. 02 x 1023 particles. Molar Mass (# grams) 6. 1490 Roth's Church Road, Spring Grove, PA 17362. 11, Solutions Name_____ MULTIPLE CHOICE. Murdoch Page 9 of 61 Website upload Key The hydrated (aqueous) ions are kept separated by the motion of the water molecules that are attracted to them. PROBLEMS Wr[te the answer on theline to left Show all your work in the space provided. 07 ÷ 2 × 22.
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2020-09-29 19:48:03
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http://www.physicsforums.com/showthread.php?t=184011
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# Average Rate of Change (word problem)
by xCanx
Tags: average, rate, word
P: 45 I'm having trouble understanding this word problem. A construction worker drops a bolt while woring on a high-rise building 320m above the ground. After t seconds, the bolt has fallen a distance of s metres, where s(t) = 320-5t^2, o < t < 8 a. Find the average velocity during the first, third and eighth seconds. I have to use f(x + h) - f(x)/h I got 5(t +h) The answer is 5m/s for the first one. How do I get the rest of them (third and eighth second)?
Math Emeritus Sci Advisor Thanks PF Gold P: 39,683 the "first second" is from t= 0 to t=1, of course: average speed during the first second is s(1)- s(0) since here h= 1. The "third second" is between t= 2 and t= 3 so average speed during the third second is s(3)- s(2) and the "eighth" second is between t= 7 and t= 8 so average speed for the eighth second is s(9)- s(8).
P: 45
Quote by HallsofIvy the "first second" is from t= 0 to t=1, of course: average speed during the first second is s(1)- s(0) since here h= 1. The "third second" is between t= 2 and t= 3 so average speed during the third second is s(3)- s(2) and the "eighth" second is between t= 7 and t= 8 so average speed for the eighth second is s(9)- s(8).
Thank you :)
so I plug in h into 5(t+h)^2????
P: 45 Average Rate of Change (word problem) this is what I have so far
P: 14 You are making the problem too difficult. Velocity is the change of position over time. The formula you list is the formula for position.
P: 3,016 $$\frac{s(t+h)-s(t)}{h}$$ is the formula for average velocity. It is a very wordy way of saying $$\frac{\Delta x}{\Delta y}$$, which is the change in y over the change in x. (<--slope) Now in order to see this, you know that $$\Delta y =y_{final}-y_{initial}$$ right? Can you see how that is the same as $$s(t+h)-s(t)$$ ? It is saying: take the position of some initial time (t) PLUS some amount h and subtract the original initial time from it. So in your second problem after 3 seconds, since your initial time is 0 and your final time is 3 then t+h means 0+3 which is of course 3. You see h is the change in time from 0 to 3. So now back to the problem; you should evaluate it as $$v_{avg}=\frac{s(3+0)-s(o)}{3}$$
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2014-09-19 09:53:50
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http://mathoverflow.net/questions/143882/how-many-n-2-cycles-can-a-cubic-graph-have
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# How many n/2-cycles can a cubic graph have
Given a simple cubic graph with $n$ vertices (which implies that $n$ is even), what is a good upper bound on the number of cycles of length $n/2$ it can have?
A random cubic graph has $\Theta((4/3)^n/n)$ cycles of length $n/2$, if I did my sums right. So do random cubic bipartite graphs. Also the whole cycle space has size $2^{n/2+1}$, so twice that is a (silly) upper bound.
Where's the truth?
ADDED: Counts for n=4,6,...,24: 0,2,6,12,20,20,48,48,132,118,312 (not in OEIS). All these are unique except that for 20 vertices there are two graphs.
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How about (3n/2) choose (n/2)? Another silly bound, but it might lead somewhere. You could also divide the graph into connected groups of 5 edges and note that a cycle can intersect each group of edges in only 8 ways. (I assume each group is acyclic, perhaps a bad move.) Gerhard "Spins Me Round Right Round" Paseman, 2013.10.03 – Gerhard Paseman Oct 3 '13 at 18:16
Interesting question! I communicated it to Michael Krivelevich. He does not know, but suggests that the methods from ams.org/mathscinet-getitem?mr=2520275 might be useful – Boris Bukh Oct 3 '13 at 18:20
The related integer sequence appears to be $4,12,24,40,40,96,96$ – Jernej Oct 3 '13 at 23:21
@Jernej: Agreed if you count oriented cycles. I'm posting counts for unoriented cycles. – Brendan McKay Oct 4 '13 at 2:02
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2016-02-13 13:01:48
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https://math.stackexchange.com/questions/3394608/finding-sum-of-digits-of-a-number-in-an-olympiad-question
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# Finding sum of digits of a number in an olympiad question.
The question is question no $$6$$ from 2016 IWYMIC olympiad individual round (It can be found here.)
Rephrased in my words , in says :
Let $$A =\,\underbrace{ 6666...66666} _{2016 \, times}\,\,$$ and $$\,\,B =\,\underbrace{ 99999...99999} _{2016 \, times}$$
Let $$N = A\times B.$$ Find the sum of digits of $$N$$.
My Approach :
Using simple observation , we can see that :
$$6\times 9 = 54 \quad \quad \rightarrow S(N) = 9$$ $$66\times 99 = 6534 \quad \quad \rightarrow S(N) = 18$$ $$666\times 999 = 665334 \quad \quad \rightarrow S(N) = 27$$ Similarly we can conclude that:
$$\,\underbrace{ 6666...66666} _{2016 \, times}\,\, \times\,\,\,\underbrace{ 99999...99999} _{2016 \, times}\quad \quad \rightarrow S(N) = 2016\times 9 = 18144$$
Which is indeed the correct answer. But how can we mathematically prove this is always correct ?
My Take :
$$\underbrace{ 6666...66666} _{2016 \, times} = 6\, (\underbrace{ 1111...11111} _{2016 \, times})$$
And
$$\underbrace{ 9999...99999} _{2016 \, times} = 9\, (\underbrace{ 1111...11111} _{2016 \, times})$$
Therefore
$$N = 54 \,\,(\underbrace{ 1111...11111} _{2016 \, times})^2$$
And then Find A pattern in $$(\underbrace{ 1111...11111} _{2016 \, times})^2$$, Although the obvious pattern is only for small values and I lost my way quickly.
Can anyone suggest me a hint to prove this?
Suggestion: Rather than factoring out the $$9$$, note that $$999\cdots99 = 10^n-1$$. This makes it not too difficult to show that $$666\cdots 66\times 999\cdots99 = 666\cdots 665333\cdots 334$$ and summing the digits here is easy.
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2022-10-07 22:03:17
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http://mlwiki.org/index.php/Inversion_Count
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ML Wiki
Inversion Count
Sequence inversion
• In a sequence $\pi = \langle a_0, ..., a_t \rangle$ or elements $A = \{ a_i \}$
• a pair $(a_i, a_j)$ is an inversion if $i < j \land a_i > a_j$
• the number of such inversions is the inversion number of sequence $\pi$
• this is a measure of "sortedness" of sequence $\pi$
Two ranked vectors
• An inversion in two rankings $r_1, r_2$ of the same variable $X$ is
• a pair $(x_i, x_j) \ | \ r_1(x_i) < r_1(x_j) \land r_2(x_i) > r_2(x_j)$
• it's called a pair-wise disagreement between two ranking lists
Graphical Counting
we can represent two rankings as a Bipartite Graph $G = \langle N, S, E \rangle$
• $N = r_1(X)$ and $E = r_2(X)$ being two disjoint set of nodes
• $X$ is some variable, and $r_1$ and $r_2$ are different rankings of this variable
• $E$ is set of edges $E = \Big\{ \big(r_1(x), r_2(x) \big) \Big\}$ i.e. corresponding elements of $X$ are connected in this graph
Counting:
• bilayer drawing of $G$ is when there are two parallel lines, edges of $N$ are drawn on one, and edges of $S$ are drawn on another
• bilayer cross count is a pairwise intersections edges of $N$ and $S$
• bilayer cross count corresponds to the number of inversions when $N$ and $S$ are ranking vectors
Example:
• $X = \{ A, B, C, D, E \}$
• two ranking $r_1 = \langle E, B, A, C, D \rangle$ and $r_2 = \langle B, E, C, D, A \rangle$
• draw this is a bipartite graph and count the number of intersections
• so there are 3 inversions in these two rankings
Algorithms
A modification of Merge Sort can compute the # of inversions in $O(|N| \log |N|)$
|
2022-12-06 05:03:09
|
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|
https://www.prepanywhere.com/prep/textbooks/functions-11-nelson/chapters/chapter-3-quadratic-functions/materials/mid-chapter-review
|
Mid Chapter Review
Chapter
Chapter 3
Section
Mid Chapter Review
Solutions 29 Videos
For the table, calculate the second differences. Determine whether the function is quadratic.
Q1a
For the table, calculate the second differences. Determine whether the function is quadratic.
Q1b
Graph
\displaystyle f(x) = -3(x - 2)^2 + 5
Q2a
Graph
\displaystyle f(x) = 2(x + 4)(x -6)
Q2b
State the vertex, the equation of the axis of symmetry, and the domain and range.
a) \displaystyle f(x) = -3(x - 2)^2 + 5
b) \displaystyle f(x) = 2(x + 4)(x -6)
Q3
Express each function in standard form.
a) \displaystyle f(x) = -3(x - 2)^2 + 5
b) \displaystyle f(x) = 2(x + 4)(x -6)
Q4
Determine the maximum or minimum value of
\displaystyle f(x) =x^2 -6x+2
Q5a
Determine the maximum or minimum value of
\displaystyle f(x) = 2(x-4)(x+ 6)
Q5b
Determine the maximum or minimum value of
\displaystyle f(x) = -2x^2 + 10x
Q5c
Determine the maximum or minimum value of
\displaystyle f(x) = -2x^2 + 10x
Q5d
The profit function for a business is given by the equation P(x) = -4x^2 + 16x -7, where x is the number of items sold, in thousands, and P(x) is dollars in thousands. Calculate the maximum profit and how many items must be sold to achieve it.
Q6
The cost per hour of funning an assembly line in a manufacturing plant is a function of the number of items produced per hour. The cost function is C(x) = 0.3x^2 -1.2x +2, where C(x) is the cost per hour in thousands of dollars, and x is the number of items produced per hour, in thousands. Determine the most economical production level.
Q7
The sum of two numbers is 16. What is the largest possible product between these numbers?
1.19mins
Q8
a) Determine the equation of the inverse of the quadratic functions f(x) = x^2 -4x + 3.
b) List the domain and range of f(x) and its inverse.
c )Sketch the graph of f(x) and its inverse.
Q9
The revenue for a business is modelled by the function R(x) = -2,8(x -10)^2 + 15, where x is the number of items sold, in thousands, and R(x) is the revenue in thousands of dollars. Express the number sold in terms of the revenue.
Q10
Almost all linear function have an inverse that is a function, but quadratic functions do not. Explain why?
Q11
Graph f(x) = - \sqrt{x + 3} and determine
a) the domain and range of f(x)
b) the equation f^{-1}
Q12
Simplify.
\displaystyle \sqrt{48}
Q13a
Simplify.
\displaystyle \sqrt{68}
Q13b
Simplify.
\displaystyle \sqrt{180}
Q13c
Simplify.
\displaystyle -3\sqrt{75}
Q13d
Simplify.
\displaystyle 5\sqrt{98}
Q13e
Simplify.
\displaystyle -9\sqrt{12}
Q13f
Simplify.
\displaystyle \sqrt{7} \times \sqrt{14}
Q14a
Simplify.
\displaystyle 3\sqrt{5} \times 2\sqrt{15}
Q14b
Simplify.
\displaystyle \sqrt{12} + 2\sqrt{48}-5\sqrt{27}
Q14c
Simplify.
\displaystyle 3\sqrt{28} - 2\sqrt{50} + \sqrt{63} -3\sqrt{18}
0.40mins
Q14d
Simplify.
\displaystyle (4- \sqrt{3})(5+2\sqrt{3})
\displaystyle (3\sqrt{5} + 2\sqrt{10})(-2\sqrt{5} + 5\sqrt{10})
|
2021-03-01 10:08:14
|
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|
http://ringtheory.herokuapp.com/keywords/keyword/15/
|
# Keyword: matrix ring
Description: Given a ring $R$ and a natural number $n$, the set $M_n(R)$ with ordinary matrix operations.
|
2018-11-19 06:12:25
|
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|
https://zbmath.org/?q=an%3A1092.68676
|
## Approximate entropy reducts.(English)Zbl 1092.68676
Summary: We use information entropy measure to extend the rough set based notion of a reduct. We introduce the Approximate Entropy Reduction Principle (AERP). It states that any simplification (reduction of attributes) in the decision model, which approximately preserves its conditional entropy (the measure of inconsistency of defining decision by conditional attributes) should be performed to decrease its prior entropy (the measure of the model’s complexity). We show NP-hardness of optimization tasks concerning application of various modifications of AERP to data analysis.
### MSC:
68T37 Reasoning under uncertainty in the context of artificial intelligence 94A17 Measures of information, entropy
|
2022-08-15 07:21:47
|
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|
https://www.lmfdb.org/EllipticCurve/Q/15680/bz/
|
# Properties
Label 15680.bz Number of curves 4 Conductor 15680 CM no Rank 1 Graph
# Related objects
Show commands for: SageMath
sage: E = EllipticCurve("15680.bz1")
sage: E.isogeny_class()
## Elliptic curves in class 15680.bz
sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
15680.bz1 15680bm3 [0, 0, 0, -20972, 1168944] [2] 18432
15680.bz2 15680bm2 [0, 0, 0, -1372, 16464] [2, 2] 9216
15680.bz3 15680bm1 [0, 0, 0, -392, -2744] [2] 4608 $$\Gamma_0(N)$$-optimal
15680.bz4 15680bm4 [0, 0, 0, 2548, 93296] [2] 18432
## Rank
sage: E.rank()
The elliptic curves in class 15680.bz have rank $$1$$.
## Modular form 15680.2.a.bz
sage: E.q_eigenform(10)
$$q + q^{5} - 3q^{9} - 4q^{11} - 2q^{13} - 2q^{17} + 4q^{19} + O(q^{20})$$
## Isogeny matrix
sage: E.isogeny_class().matrix()
The $$i,j$$ entry is the smallest degree of a cyclic isogeny between the $$i$$-th and $$j$$-th curve in the isogeny class, in the LMFDB numbering.
$$\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)$$
## Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
|
2020-02-27 15:49:58
|
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|
http://www.program-transformation.org/Stratego/StrategoRelease015
|
Page
Web
Wiki
# Stratego/XT 0.15
Stratego -- Strategies for Program Transformation
Stratego/XT 0.15 -- released July 5, 2005
This is an experimental release featuring a refactoring of the internals of the language definition and compiler. Please report problems to [email protected]
See the installation instructions if you are not familiar with the standard installation procedure of tarballs or RPMs.
Source tar.gz
Source RPM
Redhat Linux RPM
Redhat 9.0:
SuSE Linux RPM
SuSE 9.0:
Fedora Core RPM
Fedora Core 2:
Fedora Core 3:
• Build of rpm on FC3 failed because of a bug in GCC
Microsoft Windows Cygwin binaries
Sorry, no Cygwin binaries for this release.
Mac OS X binaries
Sorry, no OS X binaries for this release.
Nix:
StrategoXT is free software; you can redistribute it and/or modify it under the terms of the GNU Lesser General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version.
This software is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more details.
### Support
Despite the disclaimer above we do our best to help users of Stratego/XT. Subscribe to the Stratego mailing lists, in particular the stratego-announce and stratego mailing lists to get announcements of new releases and ask questions about usage of the languages and tools. Also we're interested to know what people are using Stratego/XT for and how it might be improved, so feel free to drop us a line.
## Summary of Changes
This is a release of a major milestone in the Stratego Core refactoring project. It is not considered to be a production release to be used by average users. Use of the release for testing by seasoned Stratego developers who maintain applications in the Stratego/XT environment, would be appreciated, however.
This release will be followed by a number of minor releases, introducing improvements in the areas of the compiler back-end, dynamic rules, the module system, and the optimizer. This should result in a stable and much improved Stratego compiler in release 0.16.
Since the distributions before 0.16 are experimental, distributions are only provided through the Nix release management system. No binaries for Windows and MacOSX? will be available. RPMs for Fedora Core 3 are not provided due to a bug in GCC in that distribution.
## Stratego Language
### Stratego Core Language
The Stratego Core language project constitutes a major refactoring of the syntax definition of the language, also requiring a refactoring of all tools and components processing Stratego programs (in particular the compiler).
The project was prompted by the problems encountered in building the first version of the Stratego Optimizer in 2003. In particular, with the representation of term annotations in the abstract syntax. In addition the language design had eroded somewhat over the years through the addition of new features.
Thus, the aims of the project are threefold:
(1) Clearly distinguish a core language to be used as intermediate language in the compiler.
(2) Explicitly represent term annotations in the abstract syntax of the core language.
(3) Cleaning up the language, removing constructs that have not proven to be useful, or have been subsumed by more more general constructs.
### Impact
Despite the changes in the structure of the language, the syntax is backwards compatible. This entails that existing applications should still work. Except of course for applications using constructs that have been removed. However, the decision to remove constructs was based on the fact that these were not used (some not even document), so this shouldn't pose a big problem. The only change with potential impact is discontinuation of support for old style dynamic rules. New style dynamic rules are the default in StrategoXT 0.14 and should be a good platform for preparing migration.
### Core vs Sugar
The syntax definition is now divided into a Stratego-Core language that is extended to the full language in Stratego-Sugar, adding syntactic abstractions to the Core. The Core is a strict subset of the full language. In addition, a number of regular tree grammars has been defined that denote a number of intermediate languages between Stratego-Sugar and Stratego-Core. These RTGs are used by the compiler to verify the sanity of the compiler components.
### Term Annotations
Term annotations were not originally supported in Stratego. The addition was handled by desugaring the matching and building of terms with annotions to calls to ATerm library functions using primitives in an early stage of compilation. The implicit presence of annotations made transformations within the compiler very fragile, and entailed that certain optimizations could not be defined properly.
In the new core syntax, terms always have an annotation. A pre-term (pt) has the form
pt := i | r | x | c(t1,...,tn)
and denotes a term without annotation. A term (t) is a pre-term with an annotation (which is itself a pre-term):
t := pt{^pt}
This is the explicit representation of term annotations; every term always has a list of annotations. That is, the annotation pre-term should always be a list of the form
Cons(t1,...,Cons(tn, Nil())
Since most terms do not have annotations, or we're not interested in them, in the sugared extension of the core language, term annotations are optional. Thus, in Stratego-Sugar, the syntax of terms is extended with:
t := pt | pt{t1,...,tn}
The interpretation of this syntax depends on the use of the term in a match or build position. That is, the following desugaring rules apply:
|[ !pt ]| -> |[ !pt{^Nil()} ]|
|[ !pt{t1,...,tn} ]| -> |[ !pt{^Cons(t1,...,Cons(tn,Nil()))} ]|
|[ ?pt ]| -> |[ ?pt{^_} ]|
|[ ?pt{t1,...,tn} ]| -> |[ ?pt{^Cons(t1,...,Cons(tn,Nil()))} ]|
(Note that this is a bit simplified, as the Conses and Nils in the right-hand side have annotations themselves, except for the outermost ones.)
So a rewrite rule
A : Plus(x, y) -> Plus(y, x)
is desugared to
A : Plus(x, y){^_} -> Plus(y, x){^Nil()}
(Eelco Visser)
### Disambiguation
The syntax of types of higher-order arguments of strategy definitions was ambiguous. Thus, the type of the skip argument in
topdownS(s : ATerm() -> ATerm() , skip : a -> a * a -> a) = ...
could be parsed as
a -> (a * a -> a) or ((a -> a) * a) -> a
This has been solved by only allowing function types as argument types when between parentheses. Thus, the above is not syntactically valid, but should now be written as
topdownS(s : ATerm() -> ATerm() , skip : (a -> a) * a -> a) = ...
It is better to require explicit disambiguation since it is easy to have the wrong expectation from implicit disambiguation in this case.
There was no priority definition between guarded left-choice and non-deterministic choice, such that the expression
s1 < s2 + s3 + s4
could be parsed in two ways, i.e.,
s1 < (s2 + s3) + s4 or s1 < s2 + (s3 + s4)
This has been resolved by giving the + operator higher priority than guarded choice to prefer the latter over the former. The first variant can be selected by using explicit parentheses.
(Eelco Visser)
### Rules in let
Let bindings can now introduce new rules. For example,
main =
let Foo : 1 -> 2
in <Foo> 1 => 2
end
Local rules shadow rules with the same name (and arity) at top-level. The scope of the rule variables is the enclosing definition, not just the rule itself! This is in line with the semantics of local definitions, but the practicality of this design choice should be reviewed.
(Martin Bravenboer)
### Outdated constructs no longer supported
A number of language constructs have been subsumed by more general constructs, or were simply never used. These constructs have been removed from the language definition starting with this release.
Old style dynamic rules are no longer supported. Everything expressible with old style rules can also be expressed with new style dynamic rules. See the paper 'Program Transformation with Scoped Dynamic Rewrite Rules' for definition and examples of new style dynamic rules.
Contextual rules can now better be expressed using dynamic rules. Local uses of contextual rules can also be defined using a local traversal. See the paper 'Strategic Pattern Matching' for a discussion of translation of contextual rules that can be applied by hand.
\begin{code}
foo = ...
\end{code}
are no longer supported. Use standard comment delimiters instead.
Other constructs that are discontinued are strategy rules of the form
L :: s1 --> s2
The thread traversal operator (can be expressed in the library), and threading and distributing congruence operators.
(Eelco Visser)
## Stratego Core Compiler
The Stratego compiler now compiles programs according to the Stratego-Sugar syntax definition. The front-end of the compiler is drastically refactored and simplified. The optimizer has been disabled in this release. Migration and improvement of the optimizer has been planned for release 0.15.4.
### Format Checking
The compiler monitors its own integrity by checking the format of intermediate results against subsets of the Regular Tree Grammar of the complete syntax definition. By default checks are only carried out at a few places, to save compile time. In case an error occurs, the level of checking can be increased using the --format-check option of strc.
(Eelco Visser)
### List Variables
The treatment of list variables is cleaned up by improving their assimilation in meta-explode. This has made it possible to have fewer compiler components be aware of list variables. If list variables in concrete syntax quotations should be used as list variables outside those quotations as well, they should have a * suffix. See issue STR-321.
Realization of this clean-up required a bugfix in implode-asfix and a change in the assimilation of concrete syntax, which entails that Conc(ts1,ts2) is now used as a special constructor to denote the concatenation of two lists. In particular, if Conc(ts1,ts2) is used in a build, this is interpreted as (ts1, ts2). In other words Conc, cannot be used as a normal constructor.
(Eelco Visser)
## Stratego Library
The definition of collect-all with a skip has been adapted to recurse to the current term instead of the children of current term in the case of application of the skip strategy.
(Report by Ron de Bruijn, Fix by Martin Bravenboer)
The string concatenation strategy conc-strings now supports tuples of more than 2 strings.
(Rob Vermaas)
## Tools
### GPP
pptable-diff now takes the arity of constructors into account.
(Martin Bravenboer)
### Stratego-regular
The implementation of sig2rtg was not compatible with the new Stratego syntax. Since the tool does not seem to be used, migration has been deferred.
(Eelco Visser)
## StrategoXT Deployment and Build System
### Build order of stratego-front and stratego-lib
The build order of stratego-front and stratego-lib has been swapped in case of a baseline build. The modules in the library should be parsed with the local syntax definition of Stratego, instead of the syntax definition of the baseline. Since the library should be used with the compiler in the current package, it should be compatible with that compiler and use the same syntax definition. When bootstrapping (building from a pre-compiled source tarball), the build-order is reversed since the library is needed for the compilation of the components in stratego-front. Also in the case of a bootstrap build, xtc is built before stratego-front.
(Eelco Visser)
## Detailed List of Issues
The full list of issues closed in this release is available at:
The release page contains the source distributions, binary RPMs, and detailed instructions on how to install the distributions:
## Bugs and Known Problems
See our issue tracking systems for reports about (open) bugs:
Please report any problems with installation or bugs in the implementation to our issue tracking system. Please check the existing issues to see if a report about the problem was already submitted.
## Contributions
The Stratego Core refactoring was carried out by
• Eelco Visser
Other improvements, bug reports, and beta tests carried out by
• Martin Bravenboer
• Ron de Bruijn
• Rob Vermaas
|
2020-07-07 18:35:47
|
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|
https://byjus.com/physics/unit-of-torque/
|
# Unit of Torque
Torque Units
Common symbols
$\tau$, M
SI unit
N⋅m
In SI base units
kg⋅m2⋅s−2
Other units
pound-force-feet, ozf⋅in, lbf⋅inch
Dimension
ML2T−2
Before we dive into the unit of torque we need to understand what torque is and a few related things. Having said that, if we recall learning about force, it is basically a form of energy that causes an object to move in a certain direction and to a certain distance. Likewise, when we talk about torque it is nothing but the measure of the force that makes an object rotate on an axis and acquire angular acceleration. Torque is classified as a vector quantity and is typically represented by the symbol which is a Greek letter tau in lowercase.
Now, let’s look at some of the units that are used to measure torque.
## SI Unit of Torque
The SI unit for torque is the Newton-meter or kgm2sec-2. How we have come to this? If we look at the formula Torque = Force X Distance. While distance is measured in meters and force is measured in newton so torque is measured in newton ⋅ meter.
However, students should remember one main point which is energy and torque are not the same concepts. Therefore, we use different unit names which is newton metres for torque and joules for energy. This is done to avoid confusions and misunderstandings.
### Other Torque Units
Torque is also sometimes measured in imperial units like inch-pounds-force, pound-force-feet (lbf⋅ft), ounce-force-inches (ozf⋅in) and foot-pounds-force. Some of the other non-SI units of torque include metre-kilograms-force.
Related articles:
Stay tuned with BYJU’S to learn various interesting physics topics with the help of interactive video lessons.
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2020-05-31 07:35:52
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https://brilliant.org/problems/again-a-good-problem/
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# This can't be that Short!
Algebra Level 4
$\displaystyle \sum_{k=1}^m \left [ (k^2 + 1) \ k! \right ] = 1999 \times 2000!$
What value of $$m$$ satisfies the above summation?
×
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2017-10-23 15:42:33
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https://cs-syd.eu/posts/2020-07-28-how-to-deal-with-money-in-software
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# How to deal with money in software
Date 2020-07-28
I recently did my accounting for my investment account. I noticed a significant amount of errors in the account statement that my broker gave me.
After a bit of a rant on Twitter, I want to explain what they likely did wrong and how to deal with money correctly in software.
## The issue
While doing my accounting, I noticed about ten mistakes of the following kind over a timespan of less than two years:
• $4999.10 - 0.50$ is $4998.59$ apparently:
• $174.13 - 0.02$ is $174.12$ apparently:
• $292.44 - 0.03$ is $292.42$ apparently:
## Disclaimer
I have contacted my broker about this privately, and asked them to confirm that they are not using IEEE754 floating point numbers for calculations involving money. They refused to confirm that they are not using IEEE754 floating point numbers for calculations involving money. They have also explicitly stated that "All entries on your account are rounded to the nearest cent." and they have made it clear that they do not see this as a problem. I have also offered my services, as someone who has experience with getting things like this exactly right, but I did not get a response to that offer.
If you have only come here to get outraged about how my broker does not care about the correctness of their software and their accounting, you can stop reading now. The rest of this blogpost will be a detailed explanation of the problems at hand and how to solve them.
## Problems with money
Because computers are really good at doing computations, you might think that they must therefore be good at dealing with money. However, money is a very complex subject. There are all sorts of pitfalls, and this list of falsehoods is a good place to start learning about them.
### Money as a floating point value
If you take away one thing from this blogpost, let it be this:
You must NOT store monetary values as IEEE 754 floating point numbers.
Corollary: Given that this is the only numeric format that JavaScript has, this implies that you must not use JavaScript for calculations with currency.
These are the primary reasons:
1. Calculations involving floating point numbers usually have some amount of error.
2. Many floating point numbers do not make sense as a value of money.
### Creating money out of thin air using floating point calculations
Floating point number calculations usually have some error. As an example, let us have a look at the value $0.1$, which cannot represented accurately using an IEEE754 floating point number. You can tell this by looking at 0.1 + 0.1 + 0.1:
λ> 0.1 + 0.1 + 0.1
0.30000000000000004
This means that when you get a dime from a customer three times, and store it as a floating number, you will have effectively created 0.00000000000000004 USD out of thin air. This might not seem like much (even though it does not need to be much, it is still wrong), but this error grows as the values in your calculations grow. There is an entire field of mathematics dedicated to thinking about what happens to the errors depending on what types of calculations you are performing.
### Representing nonsense values of money
The IEEE754 Floating point number spec defines NaN, +Infinity and -Infinity. These make no sense when interpreting them as monetary values. I have to mention this because these values form a real problem in practice.
You might be thinking: OK, but I take good care not to run into issues with these non-normalised floating point values. You are still wrong, but let us assume you are not. You will still run into the next problem
#### Minimal quantisations
The smallest denomination of US dollar that you can hold or transact is 0.01 USD: a penny. Any amount of US dollars more granular than that does not exist in coins and cannot usually be transacted. (You can still potentially have more granular debts, as you will see below.)
Even worse: different currencies have different minimal quantisations. In Switzerland, where I am using my broker, the smallest coin of a Swiss Frank is 5 "rappen": 0.05 CHF. The transaction granularity goes down to 0.01 CHF in bank transactions but no lower than that either.
## Solutions
### Requirements
In order to begin talking about solutions, let us clarify what needs to be possible in the system that we produce:
1. We must be able to add up amounts of money.
This is useful to keep a running balance, for example.
The additions must happen in such a way that the representation of the sum equals the sum of the representations. Otherwise we might create or destroy money via addition.
For example, 0.1 + 0.1 + 0.1 must somehow equal 0.3 and not 0.30000000000000004.
#### Integral Multiplication
1. We must be able to multiply amounts of money by an integer.
This is useful to find out how much multiple units of an item would cost, in the case of integral multiples.
You will notice that if you can solve the problem of addition, then you have also solved the problem of correct integral multiplication However, ideally, an integral multiplication would take up the same amount of time as a single addition, not a linear multiple of it. We would like to have multiplication be efficient as well as correct.
For example, 3.33 x 3 must equal 9.99 and not 10.
#### Division by an integer
1. We must be able to divide an amount of money by an integer.
This is useful for dividing a payment into installments. Here it is important that the sum of the installments equals the amount that was divided.
For example, 10 / 3 must be divided into three times 3.33 and an extra 0.01 somewhere: 3.34 + 3.33 + 3.33.
#### Fractional Multiplication
1. We must be able to multiply amounts of money by a fraction.
This is useful for calculating interest rates, taxes and dividends, for example.
It should happen in such a way that the representation of the product equals the product of the representations. Failing that (because we'll see that this is impossible), it should at least not create or destroy money.
For example, 0.1 times 0.11 USD, using a minimal quantisation of 0.01 USD, should result in 0.01 USD. The representational error which mathematically equals 0.001 should be dealt with somehow.
#### Accurate Accounting
1. We must be able to accurately account for every smallest unit of currency.
When earning or spending money, we must be able to account for every smallest unit of currency in play. No such units be created or destroyed, and no amounts smaller than the smallest unit may be accounted for.
Note that you can choose a different minimal quantisation of currency in your accounting system than is used for bank transactions. Even if banks use 0.01 CHF as a minimal transaction value, you can still use 0.0001 CHF as a minimal quantisation internally. The only real constraints are correctness: you do not create or destroy money and storage size. Dealing with more precise numbers requires more storage.
When you choose a different minimal quantisation of currency internally than the one you show to users, you have to somehow make that conversion. When you do, it is important that you don't add any mistakes back in.
This is likely what my broker did wrong. They probably used rounding from 4 decimals (a common standard) to 2 decimals and introduced rounding errors in the process.
#### Granular pricing
1. We must be able to make calculations with values smaller than the smallest unit of currency
The utility of this requirement may not be obvious. However, when dealing with an item of large volume and cost such as oil, we want to be able to reason about (and compete on) the (average) price of a small amount of it. In such a case it may be necessary to reason about currency in units smaller than its minimal quantisation.
For example, if I want to rent out digital storage for $0.000023$ USD per kilobyte per month, that rate is like expressed in a more precise granularity then your system deals with. In that case we must still be able to rent out $2'000'000$ kilobyte for a month for 46.00 USD.
### Representation
It is already clear that we must not represent currency as floating point numbers. Instead, we must use a representation with infinite precision: integers.
For the purposes of all but the special case of 'granular pricing', we must use integers to represent currency. For each individual currency we will represent an amount of it as an integral number of its smallest units.
For example, 1 USD could be represented as 100 at a minimal quantisation of 0.01 USD because 100 x 0.01 USD = 1 USD and 1 CHF could be represented as 20 at a minimal quantisation of 0.05 CHF because 20 x 0.05 CHF = 1 CHF. If you choose a smaller minimal quantisation, such as 0.0001 CHF, then you would represent 1 CHF as 1000 instead. Given that the minimal quantisation is not obvious, make sure to document it well (and/or put it on the type-level).
In an ideal world you would use arbitrarily-sized integers, but often you may be forced to use bounded representations with fixed limits. It is possible to accommodate such constraints but the trade-offs, as you will read next, are real and should be considered carefully. In what follows we will continue by assuming you are using (bounded-size) integer numbers.
### Addition and integer multiplication
Addition of integers is relatively simple and well supported. However, most implementations of addition are not suitable for use with fixed-sized integer representations. Indeed, most implementations of addition will just roll over around the bounds.
For example, consider a situation in which you are a bank and you store amounts of money as int32. A very wealthy client of yours has about twenty million dollars. You save this information as "this client has two billion cents". A bit later, he earns another two million dollars, but when the money is sent to him, suddenly the system says that he is millions in of dollars in debt. There are two big issues with this scenario: 1. The client has lost money and 2. The total amount of money has decreased:
λ> 2000000000 + 200000000 :: Int32
-2094967296
Instead, we must use an implementation of addition that can deal with overflow safely (see my recommendations below).
### Integer division
When dividing an amount of money by an integer, we have the problem of remainders.
As an example, let us look at a payment of 1000.00 CHF that needs to be paid in three installments. Let's assume we chose a minimal quantisation of 0.05 CHF. Mathematically dividing $1000.00$ by $3$ yields $333.33..$ repeated. However, 0.03 CHF does not exist when using this minimal quantisation and repeating fractions definitely do not, so what do we do?
In practice, we should round down the result to the nearest existing amount of currency. In this case that is 333.30 CHF. Then we need to calculate the remainder, which is 1000.00 CHF - 3 x 333.30 CHF = 0.10 CHF and decide what to do with it.
You could assign the remainder to any of the installments to get 333.40 CHF, 333.30 CHF and 333.30 CHF. You could also divide the remainder among the installments to get 333.35 CHF, 333.35 CHF and 333.30 CHF. This part is up to you, but as long as you deal with the remainder, you should end up with accurate accounting.
### Fractional multiplication, accurate accounting and granular pricing
As long as we only perform addition and integer multiplication, the accurate accounting requirement is already fulfilled. It also turns out that if we can solve fractional multiplication in an accurate way, that we can get granular pricing for free.
The big issue with fractional multiplication is easily illustrated with currency conversions. For example, as of the time of writing, the conversion rate from EUR to CHF is 1 EUR = 1.072032 CHF. At a first glance this really does not make sense because 1.072032 CHF really does not mean anything if we choose a minimal quantisation greater than 0.0000001 CHF. (Note that there will always be rates that don't make sense for your minimal quantisation, so you have to deal with this problem.) There exists 1.05 CHF and 1.10 CHF but there can never be 1.072032 CHF using this minimal quantisation. However, you never really convert exactly one EUR either. This is the important piece of info that will allow us to solve the problem.
You cannot simply multiply 1 EUR by 1.072032 CHF/EUR to get a sensible value in CHF. Instead, a bit more infrastructure is needed. The way you should interpret such a ratio is a bit different from your intuition.
To exchange 10 000 EUR to CHF, we first make a hypothetical calculation of what would happen if currencies had no minimal quantisation:
10 000.00 EUR * 1.072032 CHF / EUR = 10 720.32 CHF
Next, we choose the closest meaningful value to the result. Recall that we chose a minimal quantisation of 0.05 CHF, so the closest meaningful value is 10 720.30 CHF. We represent this as the integer 214406 because 214406 * 0.05 CHF = 10 720.30 CHF.
You now change the rate that you give so that this matches nicely, and represent the rate as an integral fraction:
214 406 % 10 000 00 = 107 203 % 5 000 00
We represent only the conversion rate as a fraction, not amounts of money. Note that you have to represent this as two numbers in memory, because there is no guarantee that the result would not have unrepresentable repeating digits. Note also that the original rate only differs from this rate by 0.000002, and you can charge a higher rate if this is a problem for you. The 'error' also shrinks as transactions get bigger and the absolute (theoretical) error is never bigger than your chosen minimal quantisation so this is really no big deal in practice.
You can then safely use this rate to accurately convert 10 000.00 EUR to 10 720.32 CHF.
### Showing amounts to users
When you chose amounts to users, we need to make sure that they make sense. It should not look like we made any errors.
By far the simplest way to deal with this problem is to show amounts to users at the same level of granularity as we are storing them. Otherwise you may get errors like the ones my broker is showing me. You can still give users the option to show amounts in a more conventional way if you do not really care about these apparent mistakes.
## Libraries
I recommend safe-decimal for the purpose of storing amounts in Haskell. It allows you to force yourself to deal with arithmetic exceptions safely and choose an underlying integer type. It also deals with integer multiplication efficiently, as we had hoped for.
I cannot recommend safe-money as is because:
• It only deals with arbitrarily-sized integers.
• It has a concept of 'dense' currencies, which is misleading.
However, the idea of putting the currency and the minimal quantisation on the type-level is potentially great.
## Conclusions
It is possible to deal with currency accurately, but it is not easy. Listen to experts on this matter, and test your implementation well. You can use property-based testing, and validity-based testing in particular, to help you find problems with your implementation.
Self-management with Smos: Archiving
Know a technical team that could use strong technical leadership?
Introduction Self-management with Smos: the waiting-action report
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2022-11-26 18:16:40
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https://brilliant.org/discussions/thread/needs-solution/
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×
# Needs solution
My problem Deflection needs a solution and rating.
Note by Nishant Sharma
2 years, 8 months ago
## Comments
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Also try this challenging question Challenges in mechanics by Ronak Agarwal(Part1) · 2 years, 8 months ago
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I have posted the solution.You can check it out · 2 years, 8 months ago
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It is quite an easy question . I want to write a solution but I don't know how to upload a diagram can you please tell me how to upload a diagram.Then I will put up the solution · 2 years, 8 months ago
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You can use these sites: postimage.org, imgur.com. · 2 years, 8 months ago
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! [alt text] (the link)...delete the spaces.. · 2 years, 8 months ago
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The link of what "an uploaded image on facebook" or what · 2 years, 8 months ago
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Can you please give an example of an image upload · 2 years, 8 months ago
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upload your diagram on imgur or any other similar website. Then copy the link of that image from that website onto here. · 2 years, 8 months ago
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Ok · 2 years, 8 months ago
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@Pranav Arora , @jatin yadav , @Sudeep Salgia , @Maharnab Mitra , @David Mattingly , @Anish Puthuraya Care to read this post... · 2 years, 8 months ago
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I have posted the solution. Kindly, check it. · 2 years, 8 months ago
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2017-03-23 02:31:18
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https://testbook.com/question-answer/frac1log-_2x-frac1log-_3--602b43adedee037ad15f808d
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# $$\frac{1}{{{{\log }_2}x}} + \frac{1}{{{{\log }_3}x}} + \frac{1}{{{{\log }_4}x}} + \ldots .. + \frac{1}{{{{\log }_{50}}x}},x \ne 1$$ is equal to
This question was previously asked in
UP TGT Mathematics 2019 Official Paper
View all UP TGT Papers >
1. $$\frac{{50}}{{{{\log }_{50}}x}}$$
2. $$\frac{{49}}{{{{\log }_{49}}x}}$$
3. $$\frac{{1}}{{{{\log }_{50!}}x}}$$
4. $$\frac{{1}}{{{{\log }_{49!}}x}}$$
Option 3 : $$\frac{{1}}{{{{\log }_{50!}}x}}$$
Free
CT 1: हिन्दी (आदिकाल)
5057
10 Questions 40 Marks 10 Mins
## Detailed Solution
Concept:
We know that,
$${\log _x}y = \frac{1}{{{{\log }_y}x}}$$ ---(1)
Calculation:
Given:
$$\frac{1}{{{{\log }_2}x}} + \frac{1}{{{{\log }_3}x}} + \frac{1}{{{{\log }_4}x}} + \ldots .. + \frac{1}{{{{\log }_{50}}x}},x \ne 1$$
By using above property, It can be written as:
$${\log _x}2 + {\log _x}3 + \ldots + {\log _x}50\;\;\;\;\;\;\;\;\;\;$$
$${\log _x}\left[ {2 \times 3 \times 4 \times 5 \times \ldots \times 50} \right]$$
$${\log _x}50! \Rightarrow \frac{1}{{{{\log }_{50!}}x}}$$
Important Points
Properties of Logarithms:
1. $${\log _a}a = 1$$
2. $${\log _a}\left( {x.y} \right) = {\log _a}x + {\log _a}y$$
3. $${\log _a}\left( {\frac{x}{y}} \right) = {\log _a}x - {\log _a}y$$
4. $${\log _a}\left( {\frac{1}{x}} \right) = - {\log _a}x$$
5. $${\rm{lo}}{{\rm{g}}_a}{x^p} = p{\rm{lo}}{{\rm{g}}_a}x$$
6. $$lo{g_a}\left( x \right) = \frac{{lo{g_b}\left( x \right)}}{{lo{g_b}\left( a \right)}}$$
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2021-09-28 05:30:50
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https://proofwiki.org/wiki/Integral_Representation_of_Dirichlet_Eta_Function_in_terms_of_Gamma_Function
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# Integral Representation of Dirichlet Eta Function in terms of Gamma Function
## Theorem
$\ds \map \eta s = \frac 1 {\map \Gamma s} \int_0^\infty \frac {x^{s - 1} } {e^x + 1} \rd x$
where:
$s$ is a complex number with $\map \Re s > 0$
$\eta$ denotes the Dirichlet eta function
$\Gamma$ denotes the Gamma function.
## Proof
$\ds \int_0^\infty \frac {x^{s - 1} } {e^x + 1} \rd x$ $=$ $\ds \int_0^\infty \frac {x^{s - 1} e^{-x} } {1 - \paren {-e^{-x} } } \rd x$ $\ds$ $=$ $\ds \int_0^\infty x^{s - 1} e^{-x} \paren {\sum_{n \mathop = 0}^\infty \paren {-e^{-x} }^n} \rd x$ Sum of Infinite Geometric Sequence $\ds$ $=$ $\ds \sum_{n \mathop = 0}^\infty \paren {\paren {-1}^n \int_0^\infty x^{s - 1} e^{-\paren {n + 1} x} \rd x}$ Fubini's Theorem $\ds$ $=$ $\ds \map \Gamma s \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n} {\paren {n + 1}^s}$ Laplace Transform of Complex Power $\ds$ $=$ $\ds \map \Gamma s \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} } {n^s}$ shifting the index $\ds$ $=$ $\ds \map \Gamma s \map \eta s$ Definition of Dirichlet Eta Function
$\blacksquare$
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2023-03-29 16:31:38
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https://viewmtn.1erlei.de/wiki/TracStandalone?action=diff&version=3
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# Changes between Version 2 and Version 3 of TracStandalone
Ignore:
Timestamp:
Jun 8, 2011, 7:44:12 PM (12 years ago)
Comment:
--
### Legend:
Unmodified
v2 * Fast: Should be almost as fast as the [wiki:TracModPython mod_python] version (and much faster than the [wiki:TracCgi CGI]). * Automatic reloading: For development, Tracd can be used in ''auto_reload'' mode, which will automatically restart the server whenever you make a change to the code (in Trac itself or in a plugin). * Options for tracd: -r, --auto-reload == Cons == == Installing as a Windows Service == === Option 1 === To install as a Windows service, get the [http://www.google.com/search?q=srvany.exe SRVANY] utility and run: {{{ The spacing here is important. For Windows 7 User, srvany.exe may not be an option, so you can use [http://www.google.com/search?q=winserv.exe WINSERV] utility and run: {{{ "C:\path\to\winserv.exe" install tracd -displayname "tracd" -start auto "C:\path\to\python.exe" c:\path\to\python\scripts\tracd-script.py " net start tracd }}} === Option 2 === Use [http://trac-hacks.org/wiki/WindowsServiceScript WindowsServiceScript], available at [http://trac-hacks.org/ Trac Hacks]. Installs, removes, starts, stops, etc. your Trac service. == Using Authentication == Using tracd with Apache .htpasswd files: To create a .htpasswd file using htpasswd: {{{ sudo htpasswd -c /path/to/env/.htpasswd username tracd provides support for both Basic and Digest authentication. The default is to use Digest. ''Support for Basic authentication was added in version 0.9.'' The general format for using authentication is (replace --auth with --basic-auth if you want to use Basic auth): {{{ $tracd -p port --auth="base_project_dir,password_file_path,realm" project_path }}} where: * '''base_project_dir''': the base directory of the project specified as follows: * when serving multiple projects: ''relative'' to the project_bath * when serving only a single project (-s): the name of the project directory Don't use an absolute path here as this won't work. ''Note:'' This parameter is case-sensitive even for environments on Windows. * '''password_file_path''': path to the password file * '''realm''': the realm name (can be anything) * '''project_path''': path of the project Examples: {{{$ tracd -p 8080 \ --auth="project1,/path/to/passwordfile,mycompany.com" /path/to/project1 }}} Of course, the password file can be be shared so that it is used for more than one project: {{{ $tracd -p 8080 \ --auth="project1,/path/to/passwordfile,mycompany.com" \ --auth="project2,/path/to/passwordfile,mycompany.com" \ /path/to/project1 /path/to/project2 }}} Another way to share the password file is to specify "*" for the project name: {{{$ tracd -p 8080 \ --auth="*,/path/to/users.htdigest,mycompany.com" \ /path/to/project1 /path/to/project2 }}} === Using a htpasswd password file === This section describes how to use tracd with Apache .htpasswd files. To create a .htpasswd file use Apache's htpasswd command (see [#GeneratingPasswordsWithoutApache below] for a method to create these files without using Apache): {{{ $sudo htpasswd -c /path/to/env/.htpasswd username }}} then for additional users: {{{ sudo htpasswd /path/to/env/.htpasswd username2 }}} then for starting the tracd: {{{ tracd -p 8080 --basic-auth=environmentname,/fullpath/environmentname/.htpasswd,/fullpath/environmentname /fullpath/environmentname }}} Tracd provides support for both Basic and Digest authentication. The default is to use Digest; to use Basic authentication, replace --auth with --basic-auth in the examples below. (You must still specify a dialogic "realm", which can be an empty string by trailing the BASICAUTH with a comma.) ''Support for Basic authentication was added in version 0.9.'' The general format for using authentication is: {{{$ tracd -p port --auth=base_project_dir,password_file_path,realm project_path }}} where: * '''base_project_dir''' is the base directory of the project; note: this doesn't refer to the project name, and it is case-sensitive even for windows environments * '''password_file_path''' path of the password file * '''realm''' realm * '''project_path''' path of the project Example: {{{ $tracd -p 8080 \ --auth=project1,/path/to/users.htdigest,mycompany.com /path/to/project1 }}} Of course, the digest file can be be shared so that it is used for more than one project: {{{$ tracd -p 8080 \ --auth=project1,/path/to/users.htdigest,mycompany.com \ --auth=project2,/path/to/users.htdigest,mycompany.com \ /path/to/project1 /path/to/project2 }}} Another way to share the digest file is to specify "*" for the project name: {{{ $tracd -p 8080 \ --auth="*",/path/to/users.htdigest,mycompany.com \ /path/to/project1 /path/to/project2 }}} If using the -s parameter for serving a Trac environment from the root of a domain, one must use * for the project name == How to set up an htdigest password file ==$ sudo htpasswd /path/to/env/.htpasswd username2 }}} Then to start tracd run something like this: {{{ $tracd -p 8080 --basic-auth="projectdirname,/fullpath/environmentname/.htpasswd,realmname" /fullpath/environmentname }}} For example: {{{$ tracd -p 8080 --basic-auth="testenv,/srv/tracenv/testenv/.htpasswd,My Test Env" /srv/tracenv/testenv }}} ''Note:'' You might need to pass "-m" as a parameter to htpasswd on some platforms (OpenBSD). === Using a htdigest password file === If you have Apache available, you can use the htdigest command to generate the password file. Type 'htdigest' to get some usage instructions, or read [http://httpd.apache.org/docs/2.0/programs/htdigest.html this page] from the Apache manual to get precise instructions. You'll be prompted for a password to enter for each user that you create. For the name of the password file, you can use whatever you like, but if you use something like users.htdigest it will remind you what the file contains. As a suggestion, put it in your /conf folder along with the [TracIni trac.ini] file. Note that you can start tracd without the --auth argument, but if you click on the ''Login'' link you will get an error. == Generating Passwords Without Apache == === Generating Passwords Without Apache === If you don't have Apache available, you can use this simple Python script to generate your passwords: {{{ python trac-digest.py -u username -p password >> c:\digest.txt tracd --port 8000 --auth=proj_name,c:\digest.txt,trac c:\path\to\proj_name $python trac-digest.py -u username -p password >> c:\digest.txt$ tracd --port 8000 --auth=proj_name,c:\digest.txt,trac c:\path\to\proj_name }}} in the Wiki: [//chrome/site/software-0.1.tar.gz] The development version of Trac supports a new htdocs: TracLinks Since 0.10, Trac supports a new htdocs: TracLinks syntax for the above. With this, the example link above can be written simply htdocs:software-0.1.tar.gz.
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2022-12-06 03:32:00
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http://www.turbo-forum.com/index.php?action=recent
|
### Recent Posts
Pages: [1] 2 3 ... 10
1
##### Ridft, Rdgrad, Dscf, Grad / Re: Unrestricted DFT in H2 Dissociation
« Last post by Wook on March 27, 2017, 02:52:24 am »
Wook
2
3
##### Ridft, Rdgrad, Dscf, Grad / Re: Unrestricted DFT in H2 Dissociation
« Last post by christof.haettig on March 23, 2017, 03:03:14 pm »
I've not used Gaussian since more than 20 years.... No idea what 'guess=mix' in Gaussian does.
That symmetry has to be broken 'by hand' to get the correct UHF solution is a very special problem for higly symmetric molecules. Up to know I had this only for H2 which I use to teach students the bond breaking problem in HF and KS-DFT.
Christof
4
##### Ricc2 / Re: ADC(2) transition moment calculation
« Last post by christof.haettig on March 23, 2017, 02:58:20 pm »
I hope you do not expect me to analyze the notation from Dreuw....
The implementation in TURBOMOLE is based on the original formulation by Schirmer and coworker. They used a strictly MP-based approximation which
leads to the occurence of second-order contributions in the expressions for transition moments. That might be interesting for academic purposes,
but for a model that could be used for routine (production) calculations it is for the performance/cost ratio unacceptable to include the O(N^6) scaling
terms second-order terms.
There identification is straightforward if one notes that a) the MP2 energy is computed from the first-order amplitudes and the b) the second-order
amplitudes are those computed in a MPPT program to compute the MP4 energy.
The detailed working expressions are given e.g. by Lunkenheimer in JCTC 9 (2013) 977.
5
##### TURBOMOLE Forum General / Re: $rpacor$maxcor \$ricore questions
« Last post by inozerox on March 22, 2017, 08:10:03 pm »
Thanks christof.haettig for answering me my question.
Thinking about the answer that you give me. I have another question.
If I try to set up two calculations at the same machine. Do I need to assign the 75% of RAM memory of that machine in both cases?
6
##### Ridft, Rdgrad, Dscf, Grad / Re: Unrestricted DFT in H2 Dissociation
« Last post by Wook on March 22, 2017, 08:43:21 am »
Dear Christof
I am still wondering whether there is a way to do what you said by a command in Turbomole, like 'guess=mix' in Gaussian.
Is it possible?
Thanks,
Wook
7
##### Ricc2 / ADC(2) transition moment calculation
« Last post by david.mester on March 22, 2017, 07:55:39 am »
Dear Users and Developers,
I would like to perform ADC(2) transition moment calculations with Turbomole and make comparisons with previous results and benchmarks obtained with other programs.
Could you, please, help to clarify the precise expression for the transition moments printed by Tubromole? The corresponding section of the manual (http://www.turbomole-gmbh.com/manuals/version_6_6/Documentation_html/DOKse43.html) discusses that some terms are neglected in the ground to excited state transition moments:
“...the implementation in the ricc2 program neglects in the calculation of the ground to excited state transition moments the contributions which are second order in ground state amplitudes (i.e. contain second-order amplitudes or products of first-order amplitudes)”
It is not clear to us, which contributions are included and which ones are neglected. Could you, please, explain which terms are computed in more detail, because we were not able to find a more detailed reference/publication covering the implementation in Turbomole. Could you point us to such documentation in the literature that we perhaps missed?
To assist the discussion I collected here the terms contributing to the ADC(2) transition densities according to the Eqs. (A1)-(A3) in the paper of Dreuw and co-workers (Mol. Phys., 2014, 112, 774-784):
“0th order:
\rho_{ai} = Y_{ia}
1st order:
\rho_{ia} = - \sum_{jb} t_{ij}^{ab} Y_{jb}
2nd order:
\rho_{ij} = - \sum_{a} \rho^{MP2}_{ia} Y_{ja} - \sum_{kab} Y_{ik}^{ab} t_{jk}^{ab}
\rho_{ia} = - \sum_{jb} Y_{jb} t^D_{ij}^{ab}
\rho_{ai} = 1/2 \sum_{jb} t_{ij}^{ab} \sum_{kc} t_{jk}^{bc} Y_{kc} - 1/2 \sum_{b} \rho^{MP2}_{ab} Y_{ib} + 1/2 \sum_{j} \rho^{MP2}_{ij} Y_{ja}
\rho_{ab} = \sum_{i} Y_{ia} \rho^{MP2}_{ib} + \sum_{ijc} Y_{ij}^{ac} t_{ij}^{bc},
where Y denotes the excited state eigenvectors, t_{ij}^{ab} is the MP2 amplitude, t^D_{ij}^{ab} is an O(N^6) scaled intermediate, and \rho^{MP2}_{pq} is the pq part of the MP2 density matrix.”
Could you explain your approximation using the above terminology?
An second source for discrepancies in comparisons could come from different approaches to normalize the ground/excited state wave function. Could you explain how do you normalize these functions in Tubromole?
Yours sincerely,
Dávid Mester
8
##### Aoforce and Numforce / Re: Got two imaginary vibrational modes
« Last post by christof.haettig on March 21, 2017, 11:43:00 am »
If the structure is "wrong" or not depends on what you want to do...
9
##### Ridft, Rdgrad, Dscf, Grad / Re: Unrestricted DFT in H2 Dissociation
« Last post by christof.haettig on March 21, 2017, 11:41:17 am »
I'm not sure what you want to compute. But I have the impression that you tried to converge the SCF procedure to an excited state by asking the program to occupy for beta spin the 2nd orbital and leaving the lowest energy beta spin orbital empty. This is not supposed to work.
If you want to compute the correct (symmetry broken) spin unrestricted ground state for H2 1.) switch off symmetry and 2.) generate somehow
symmetry broken start MOs. You can do the latter by modifying by hand for one of the occupied orbitals the coefficients in the alpha or beta file
or you can do a prelimary calculation where you change the charge for one of the atoms a little bit.
For a scan of the potential curve start at large distances and then proceed to smaller distances. (Not the other way.)
10
##### Define / Re: RI-approximation in Turbomole
« Last post by christof.haettig on March 21, 2017, 11:23:53 am »
The comparison of total DFT energies between different codes is difficult, because the total energies are pretty sensitive to the integration grids
for the DFT functional and also sensitive to the details of XC functional. It would be better to compare energy differences.
Pages: [1] 2 3 ... 10
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2017-03-28 10:01:54
|
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https://mathleaks.com/study/kb/rule/relationship_between_arc_length_and_arc_measure
|
{{ item.displayTitle }}
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Rule
# Relationship between Arc Length and Arc Measure
In a circle, the ratio between an arc length and the circumference is equal to the ratio between the arc measure and the circle's angle,
### Rule
Find the arc length
This relationship can be used to calculate the length or measure of an arc.
The arc length between and can be measured using the relationship between the arc length and arc measure. The arc measure is known, and with the radius, in., the circumference can be calculated. The circumference is given by the formula Therefore, the circumference is: The values for the arc measure and the circumference are now used to find the arc length.
Solve for
Th arc length can be calculated to units.
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2021-03-02 20:27:31
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https://mathematica.stackexchange.com/questions/255577/including-literals-into-compiled-c-code?noredirect=1
|
# Including literals into compiled C code
I have a function that I want to Compile into C for speed. Inside this function is a certain long expression called x, which has been computed earlier in a Mathematica session. However, if you call x naively inside the code, then you are referring to an "external variable" and so ordinary Wolfram Language code is used instead.
See below. The second version is 100 times faster because it got properly compiled into C. How can I include x in the code without literally copying-and-pasting a huge expression, which muddies up my notebook?
x = Sin[i^2]*Cos[i] (* A fairly long complicated expression *);
compiledsum1 = Compile[{{NumPoints, _Integer}},
Block[
{i, sum = 0.0},
For[i = 0, i < NumPoints, i++, sum += x;];
sum
], CompilationTarget -> "C"];
compiledsum2 = Compile[{{NumPoints, _Integer}},
Block[
{i, sum = 0.0},
For[i = 0, i < NumPoints, i++, sum += Sin[i^2]*Cos[i];];
sum
], CompilationTarget -> "C"];
Timings:
compiledsum1[100000] // Timing
{0.841049, 223.296}
compiledsum2[100000] // Timing
0.006448, 223.296
• Sep 12, 2021 at 13:35
• Externals are inlined by "InlineExternalDefinitions" -> True, see CompilationOptions in Compile
– I.M.
Sep 12, 2021 at 22:55
x = Sin[i^2]*Cos[i] (*A fairly long complicated expression*);
compiledsumtest1 =
Hold@Compile[{{NumPoints, _Integer}},
Block[{i, sum = 0.0}, For[i = 0, i < NumPoints, i++, sum += x;];
sum], CompilationTarget -> "C"] /. OwnValues@x // ReleaseHold;
Of course the solution above isn't the simplest for your specific problem. Henrik and I.M. have already shown two simpler solutions, I'd like to add one more based on pure function:
x = Sin[i^2]*Cos[i] (*A fairly long complicated expression*);
compiledsumtest2 =
Compile[{{NumPoints, _Integer}},
Block[{i, sum = 0.0}, For[i = 0, i < NumPoints, i++, sum += #;];
sum], CompilationTarget -> "C"] &@x;
But do remember the pattern-matching-based method is more general, here's an example.
• Thank you! Can you point me to the best reference to understand all of this wizardry? The documentation is quite tricky for me to properly grasp. Sep 12, 2021 at 13:04
• @BruceBartlett You may have a look at Leonid Shifrin's book, that's where I first learned about @, OwnValues, etc. These days Stephen Wolfram's book is a good choice, too. Sep 12, 2021 at 13:17
• Thank you very much!! Sep 12, 2021 at 13:35
Even easier with With.
compiledsum3 = With[{x = x},
Compile[{{NumPoints, _Integer}},
Block[{i, sum = 0.0},
For[i = 0, i < NumPoints, i++, sum += x;];
sum
],
CompilationTarget -> "C"]
]
(This is certainly a duplicate, but I have no time to look it up...)
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2022-07-04 11:22:54
|
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https://matpitka.blogspot.com/2023/01/about-selection-of-action-defining-k.html
|
## Wednesday, January 18, 2023
### About the selection of the action defining the Kähler function of the "world of classical worlds" (WCW)
About the selection of the action defining the Kähler function of the "world of classical worlds" (WCW)
The proposal is that space-time surfaces correspond to preferred extremals of some action principle, being analogous to Bohr orbits, so that they are almost deterministic. The action for the preferred extremal would define the Kähler function of WCW (see this and this ).
How unique is the choice of the action defining WCW Kähler metric? The problem is that twistor lift strongly suggests the identification of the preferred extremals as 4-D surfaces having 4-D generalization of complex structure and that a large number of general coordinate invariant actions constructible in terms of the induced geometry have the same preferred extremals.
1. Could twistor lift fix the choice of the action uniquely?
The twistor lift of TGD (see this, this , this , and this ) generalizes the notion of induction to the level of twistor fields and leads to a proposal that the action is obtained by dimensional reduction of the action having as its preferred extremals the counterpart of twistor space of the space-time surface identified as 6-D surface in the product T(M4)× T(CP2) twistor spaces of T(M4) and T(CP2) of M4 and CP2. Only M4 and CP2 allow a twistor space with Kähler structure (see this) so that TGD would be unique. Dimensional reduction is forced by the condition that the 6-surface has S2-bundle structure characterizing twistor spaces and the base space would be the space-time surface.
1. Dimensional reduction of 6-D Kähler action implies that at the space-time level the fundamental action can be identified as the sum of Kähler action and volume term (cosmological constant). Other choices of the action do not look natural in this picture although they would have the same preferred extremals.
2. Preferred extremals are proposed to correspond to minimal surfaces with singularities such that they are also extremals of 4-D Kähler action outside the singularities. The physical analogue are soap films spanned by frames and one can localize the violation of the strict determinism and of strict holography to the frames.
3. The preferred extremal property is realized as the holomorphicity characterizing string world sheets, which generalizes to the 4-D situation. This in turn implies that the preferred extremals are the same for any general coordinate invariant action defined on the induced gauge fields and induced metric apart from possible extremals with vanishing CP2 Kähler action.
For instance, 4-D Kähler action and Weyl action as the sum of the tensor squares of the components of the Weyl tensor of CP2 representing quaternionic imaginary units constructed from the Weyl tensor of CP2 as an analog of gauge field would have the same preferred extremals and only the definition of Kähler function and therefore Kähler metric of WCW would change. One can even consider the possibility that the volume term in the 4-D action could be assigned to the tensor square of the induced metric representing a quaternionic or octonionic real unit.
Action principle does not seem to be unique. On the other hand, the WCW Kähler form and metric should be unique since its existence requires maximal isometries.
Unique action is not the only way to achieve this. One cannot exclude the possibility that the Kähler gauge potential of WCW in the complex coordinates of WCW differs only by a complex gradient of a holomorphic function for different actions so that they would give the same Kähler form for WCW. This gradient is induced by a symplectic transformation of WCW inducing a U(1) gauge transformation. The Kähler metric is the same if the symplectic transformation is an isometry.
Symplectic transformations of WCW could give rise to inequivalent representations of the theory in terms of action at space-time level. Maybe the length scale dependent coupling parameters of an effective action could be interpreted in terms of a choice of WCW Kähler function, which maximally simplifies the computations at a given scale.
1. The 6-D analogues of electroweak action and color action reducing to Kähler action in 4-D case exist. The 6-D analog of Weyl action based on the tensor representation of quaternionic imaginary units does not however exist. One could however consider the possibility that only the base space of twistor space T(M4) and T(CP2) have quaternionic structure.
2. Kähler action has a huge vacuum degeneracy, which clearly distinguishes it from other actions. The presence of the volume term removes this degeneracy. However, for minimal surfaces having CP2 projections, which are Lagrangian manifolds and therefore have a vanishing induced Kähler form, would be preferred extremals according to the proposed definition. For these 4-surfaces, the existence of the generalized complex structure is dubious.
For the electroweak action, the terms corresponding to charged weak bosons eliminate these extremals and one could argue that electroweak action or its sum with the analogue of color action, also proportional Kähler action, defines the more plausible choice. Interestingly, also the neutral part of electroweak action is proportional to Kähler action.
Twistor lift strongly suggests that also M4 has the analog of Kähler structure. M8 must be complexified by adding a commuting imaginary unit i. In the E8 subspace, the Kähler structure of E4 is defined in the standard sense and it is proposed that this generalizes to M4 allowing also generalization of the quaternionic structure. M4 Kähler structure violates Lorentz invariance but could be realized at the level of moduli space of these structures.
The minimal possibility is that the M4 Kähler form vanishes: one can have a different representation of the Kähler gauge potential for it obtained as generalization of symplectic transformations acting non-trivially in M4. The recent picture about the second quantization of spinors of M4× CP2 assumes however non-trivial Kähler structure in M4.
1. If the preferred extremals satisfy 4-D generalization of holomorphicity, a very large set of actions gives rise to the same preferred extremals unless there are some additional conditions restricting the number of preferred extremals for a given action.
2. WCW metric has an infinite number of zero modes, which appear as parameters of the metric but do not contribute to the line element. The induced Kähler form depends on these degrees of freedom. The existence of the Kähler metric requires maximal isometries, which suggests that the Kähler metric is uniquely fixed apart from a conformal scaling factor \Omega depending on zero modes. This cannot be true: galaxy and elementary particle cannot correspond to the same Kähler metric.
Number theoretical vision and the hierarchy of inclusions of HFFs associated with supersymplectic algebra actings as isometries of WcW provide equivalent realizations of the measurement resolution. This solves these paradoxes and predicts that WCW decomposes into sectors for which Kähler metrics of WCW differ in a natural way.
2.1 The hierarchy subalgebras of supersymplectic algebra implies the decomposition of WCW into sectors with different actions
Supersymplectic algebra of δ M4+× CP2 is assumed to act as isometries of WCW (see this). There are also other important algebras but these will not be discussed now.
1. The symplectic algebra A of δ M4+× CP2 has the structure of a conformal algebra in the sense that the radial conformal weights with non-negative real part, which is half integer, label the elements of the algebra have an interpretation as conformal weights.
The super symplectic algebra A has an infinite hierarchy of sub-algebras (see this) such that the conformal weights of sub-algebras An(SS) are integer multiples of the conformal weights of the entire algebra. The superconformal gauge conditions are weakened. Only the subalgebra An(SS) and the commutator [An(SS),A] annihilate the physical states. Also the corresponding classical Noether charges vanish for allowed space-time surfaces.
This weakening makes sense also for ordinary superconformal algebras and associated Kac-Moody algebras. This hierarchy can be interpreted as a hierarchy symmetry breakings, meaning that sub-algebra An(SS) acts as genuine dynamical symmetries rather than mere gauge symmetries. It is natural to assume that the super-symplectic algebra A does not affect the coupling parameters of the action.
2. The generators of A correspond to the dynamical quantum degrees of freedom and leave the induced Kähler form invariant. They affect the induced space-time metric but this effect is gravitational and very small for Einsteinian space-time surfaces with 4-D M4 projection.
The number of dynamical degrees of freedom increases with n(SS). Therefore WCW decomposes into sectors labelled by n(SS) with different numbers of dynamical degrees of freedom so that their Kähler metrics cannot be equivalent and cannot be related by a symplectic isometry. They can correspond to different actions.
2.2 Number theoretic vision implies the decomposition of WCW into sectors with different actions
The number theoretical vision leads to the same conclusion as the hierarchy of HFFs. The number theoretic vision of TGD based on M8-H duality (see this) predicts a hierarchy with levels labelled by the degrees n(P) of rational polynomials P and corresponding extensions of rationals characterized by Galois groups and by ramified primes defining p-adic length scales.
These sequences allow us to imagine several discrete coupling constant evolutions realized at the level H in terms of action whose coupling parameters depend on the number theoretic parameters.
2.2.1 Coupling constant evolution with respect to n(P)
The first coupling constant evolution would be with respect to n(P).
1. The coupling constants characterizing action could depend on the degree n(P) of the polynomial defining the space-time region by M8-H duality. The complexity of the space-time surface would increase with n(P) and new degrees of freedom would emerge as the number of the rational coefficients of P.
2. This coupling constant evolution could naturally correspond to that assignable to the inclusion hierarchy of hyperfinite factors of type II1 (HFFs). I have indeed proposed (see this) that the degree n(P) equals to the number n(braid) of braids assignable to HFF for which super symplectic algebra subalgebra An(SS) with radial conformal weights coming as n(SS)-multiples of those of entire algebra A. One would have n(P)= n(braid)=n(SS). The number of dynamical degrees of freedom increases with n which just as it increases with n(P) and n(SS).
3. The actions related to different values of n(P)=n(braid)=n(SS) cannot define the same Kähler metric since the number of allowed space-time surfaces depends on n(SS).
WCW could decompose to sub-WCWs corresponding to different actions, a kind of theory space. These theories would not be equivalent. A possible interpretation would be as a hierarchy of effective field theories.
4. Hierarchies of composite polynomials define sequences of polynomials with increasing values of n(P) such that the order of a polynomial at a given level is divided by those at the lower levels. The proposal is that the inclusion sequences of extensions are realized at quantum level as inclusion hierarchies of hyperfinite factors of type II1.
A given inclusion hierarchy corresponds to a sequence n(SS)i such that n(SS)i divides n(SS)i+1. Therefore the degree of the composite polynomials increases very rapidly. The values of n(SS)i can be chosen to be primes and these primes correspond to the degrees of so called prime polynomials (see this) so that the decompositions correspond to prime factorizations of integers. The "densest" sequence of this kind would come in powers of 2 as n(SS)i= 2i. The corresponding p-adic length scales (assignable to maximal ramified primes for given n(SS)i) are expected to increase roughly exponentially, say as 2r2i. r=1/2 would give a subset of scales 2r/2 allowed by the p-adic length scale hypothesis. These transitions would be very rare.
A theory corresponding to a given composite polynomial would contain as sub-theories the theories corresponding to lower polynomial composites. The evolution with respect to n(SS) would correspond to a sequence of phase transitions in which the action genuinely changes. For instance, color confinement could be seen as an example of this phase transition.
5. A subset of p-adic primes allowed by the p-adic length scale hypothesis p≈ 2k defining the proposed p-adic length scale hierarchy could relate to nS changing phase transition. TGD suggests a hierarchy of hadron physics corresponding to a scale hierarchy defined by Mersenne primes and their Gaussian counterparts (see this and this). Each of them would be characterized by a confinement phase transition in which nS and therefore also the action changes.
2.2.2 Coupling constant evolutions with respect to ramified primes for a given value of n(P)
For a given value of n(P), one could have coupling constant sub-evolutions with respect to the set of ramified primes of P and dimensions n=heff/h0 of algebraic extensions. The action would only change by U(1) gauge transformation induced by a symplectic isometry of WCW. Coupling parameters could change but the actions would be equivalent.
The choice of the action in an optimal manner in a given scale could be seen as a choice of the most appropriate effective field theory in which radiative corrections would be taken into account. One can interpret the possibility to use a single choice of coupling parameters in terms of quantum criticality.
The range of the p-adic length scales labelled by ramified primes and effective Planck constants heff/h0 is finite for a given value of n(SS).
The first coupling constant evolution of this kind corresponds to ramified primes defining p-adic length scales for given n(SS).
1. Ramified primes are factors of the discriminant D(P) of P, which is expressible as a product of non-vanishing root differents and reduces to a polynomial of the n coefficients of P. Ramified primes define p-adic length scales assignable to the particles in the amplitudes scattering amplitudes defined by zero energy states.
P would represent the space-time surface defining an interaction region in N--particle scattering. The N ramified primes dividing D(P) would characterize the p-adic length scales assignable to these particles. If D(P) reduces to a single ramified prime, one has elementary particle this), and the forward scattering amplitude corresponds to the propagator.
This would give rise to a multi-scale p-adic length scale evolution of the amplitudes analogous to the ordinary continuous coupling constant evolution of n-point scattering amplitudes with respect to momentum scales of the particles. This kind of evolutions extend also to evolutions with respect to n(SS).
2. physical constraints require that n(P) and the maximum size of the ramified prime of P correlate (see this).
A given rational polynomial of degree n(P) can be always transformed to a polynomial with integer coefficients. If the integer coefficients are smaller than n(P), there is an upper bound for the ramified primes. This assumption also implies that finite fields become fundamental number fields in number theoretical vision (see this).
3. p-Adic length scale hypothesis (see this) in its basic form states that there exist preferred primes p≈ 2k near some powers of 2. A more general hypothesis states that also primes near some powers of 3 possibly also other small primes are preferred physically. The challenge is to understand the origin of these preferred scales.
For polynomials P with a given degree n(P) for which discriminant D(P) is prime, there exists a maximal ramified prime. Numerical calculations suggest that the upper bound depends exponentially on n(P).
Could these maximal ramified primes satisfy the p-adic length scale hypothesis or its generalization? The maximal prime defines a fixed point of coupling constant evolution in accordance with the earlier proposal. For instance, could one think that one has p≈ 2k, k= n(SS)? Each p-adic prime would correspond to a p-adic coupling constant sub-evolution representable in terms of symplectic isometries.
Also the dimension n of the algebraic extension associated with P, which is identified in terms of effective Planck constant heff/h0=n labelling different phases of the ordinary matter behaving like dark matter, could give rise to coupling constant evolution for given n(SS). The range of allowed values of n is finite. Note however that several polynomials of a given degree can correspond to the same dimension of extension.
2.3 Number theoretic discretization of WCW and maxima of WCW Kähler function
Number theoretic approach involves a unique discretization of space-time surface and also of WCW. The question is how the points of the discretized WCW correspond to the preferred extremals.
1. The exponents of Kähler function for the maxima of Kähler function, which correspond to the universal preferred extremals, appear in the scattering amplitudes. The number theoretical approach involves a unique discretization of space-time surfaces defining the WCW coordinates of the space-time surface regarded as a point of WCW.
In (see this ) it is assumed that these WCW points appearing in the number theoretical discretization correspond to the maxima of the Kähler function. The maxima would depend on the action and would differ for ghd maxima associated with different actions unless they are not related by symplectic WCW isometry.
2. The symplectic transformations of WCW acting as isometries are assumed to be induced by the symplectic transformations of δ M4+× CP2 (see this and this). As isometries they would naturally permute the maxima with each other.
See the article Reduction of standard model structure to CP2 geometry and other key ideas of TGD or the chapter Trying to fuse the basic mathematical ideas of quantum TGD to a single coherent whole.
For a summary of earlier postings see Latest progress in TGD.
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2023-01-27 14:17:34
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https://dsp.stackexchange.com/questions/84086/effect-of-overlapping-percentage-on-stft-output
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# Effect of overlapping percentage on STFT output
I know STFT is generally applied to non-stationary signals but I tried to apply it to a stationary signal to get a working knowledge.
I created a stationary signal composed of three frequencies as below:
x = 3*cos(2*pi*30*t + phi1) + 2*cos(2*pi*45*t - phi2) + 1*cos(2*pi*70*t + phi3);
I then performed STFT (Short-Time Fourier transform) on this signal using hann window of length = 128.
1. I have tried ranging the overlapping percentage from 75% to 0% (no overlap) but cannot see and difference in the spectrogram generated. Why could that be?
2. On varying the length of the window (keeping overlapping percentage same), the bright lines in the spectrogram thickens or thins. Why could that be?
I am using MATLAB's STFT function and documentation can be found here:1
Edit:
I tried a signal with varying frequency over time. As pointed out, decreasing overlap percentage results in coarser time grid.
However, when increasing window size from 64 to 128 keeping overlap percentage same (75% for both), and 128-point FFT, in both cases the STFT is calculated for 65 frequencies (128/2 + 1). And the 64-hann window gives a better result. Does that mean that a smaller window gives better results almost everytime? Obviously, I understand that a smaller window would mean more computation cost.
Another experiment made me realize that keeping everything constant (window size, overlap), and increasing the N in N-DFT gives better results.
• Try a signal where frequency actually varies with times. Maybe an up sweep and down sweep. Aug 7 at 19:40
My answers will be intuitive, I hope. There are more rigorous mathematical arguments that can be made, but your example is not stochastic or varying in frequency, so they are not necessary here.
1. Incomplete but intuitive answer: The spectrum estimate does not appear to change with changing overlap because the spectrum is constant. But see below for more details on block-size, overlap, and time resolution versus frequency resolution.
2. If you feed a block of $$N$$ samples to an FFT (the algorithm that outputs a discrete Fourier transform (DFT)), then the output has $$N$$ "frequency bins". As $$N$$ grows, you have more frequency bins, and you have finer resolution in frequency a finer grid of frequencies. Your example has just 3 constant frequencies to estimate, so as the frequency resolution grid of frequencies becomes finer, the "weight" will be concentrated in frequency bins closer and closer to the 3 frequencies. As a result, the lines in the spectrogram will be narrower as $$N$$ increases. You can also change the window; the thickness of the lines might change as you change the window.
Note that as the block-size ($$N$$) increases, the waiting-time between feeding blocks of samples to the FFT increases. Hence, as you get greater frequency resolution a finer grid of frequencies, you have lesser time resolution a coarser grid of times. Overlapping blocks compensates for that a little bit: as the overlap increases, the waiting-time between FFTs shrinks. The cost is more computation.
I offer the diagram below to address @Lobster3321's comment on this answer. The diagram has been corrected per @OverLordGoldDragon's comment on the location of the insertion of zeros for zero-padding.
The diagram below gives a high-level view of the order of processing. If we had 0% overlap, then we would have a new column in the spectrogram after $$N$$ samples. With, 50% overlap, on the other hand, we have a new column of the spectrogram after just $$N/2$$ samples.
• It needs pointing that "resolution" in your answer does not refer to the Heisenberg sense; there only the window matters. Aug 7 at 19:36
• @OverLordGoldDragon : I was afraid that I had used an overloaded term. I will change it to something that is less ambiguous. Aug 7 at 19:38
• Crossing out what's "wrong but related" or "you might think this", instead of deleting it entirely, is an interesting and I think effective approach. Aug 8 at 11:58
• @JoeMack here, N is the windows size? Or is it the N from N-point DFT? As per the results that I'm seeing, it should be N from the N-point DFT. Please see details in edit. Would be really appreciative of your help. Aug 8 at 14:22
• The window is center-padded, not right-padded. It's also confusing, if not incorrect, to call $M$ an interpolant and $N$ "original", where I presume $M$ is n_fft and $N$ is window length? which also is inconsistent with the rest of the answer. I think there's too much focus on the "windowed DFT" interpretation of STFT; STFT is complex bandpass convolutions, it utilizes no frequencies outside of DFT(x). Aug 8 at 22:50
The DFT/FFT is a block transform, taking the inner-product of a signal vector with rows/columns of a DFT matrix. When applied in a block processing way, it makes sense to do this either back-to-back or with e.g. 50% overlap.
I find it rewarding to take this to the extreme. If you see the DFT matrix not as a block matrix multiplication, but rather as a set of convolution kernels, one can do a convolution for every sample shift of the input signal. Inner-products that are shifted one sample at a time.
With the understanding of maximally overlapped DFT processing as a bank of convolution filters, doing non-overlapped or 50% overlapped STFT is just a kind of degeneration. Namely, dropping most of the convolution results in order to reduce complexity.
Short answer but I'm posting because the other answers have correct general descriptions but are missing the point past question 1:
1. Congratulations, you've generated perfect sines and padded them correctly. Ordinarily you'd observe differences near boundaries. What's happening is, STFT of a pure sine is a perfectly horizontal line in time-frequency, since, well, it's "time vs frequency" and if the frequency never changes then it's a straight line. "Overlap" is alias for "hop size", i.e. stride of convolutions - it's identical to subsampling the STFT along time, like STFT_large_hop == STFT_small_hop[:, ::20]. If this is unclear, see top visual here, where the only difference is the window width doesn't change.
2. Changing window length by itself won't do this. Odds are, you're feeding "length" argument to a generating function like scipy.signal.windows.hanning, which necessarily generate a wider window. The thinning is a consequence of greater frequency resolution, or wider in time window: better separability along frequency, worse along time.
I also discourage the "windowed Fourier" interpretation of STFT as the sole or best one: STFT is convolution with complex bandpass filters, windowed DFT is one way to get there.
Does that mean that a smaller window gives better results almost everytime?
Narrower window is always superior for single-component signals (up to a limit that we likely won't encounter in practice) - i.e. one line you can draw in time-frequency, left-to-right, without lifting your hand. Details in this post.
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2022-09-29 16:42:21
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http://tex.stackexchange.com/tags/pgfplots/new
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# Tag Info
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You can specify the column index for each coordinate using the x index or y index keys, as in\addplot file [x index=0, y index=1] {Fre.dat};. 0 is the index of the first column, as explained in the pgfplots manual section 4.3.2. Example (removed everything not essential to the problem at hand): \documentclass[tikz,border=5pt]{standalone} ...
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The \tiny stuff in your plots has the same size in both figures -- as well as all the other text in the plots. As you mentioned already by yourself in your question it is a very bad idea to scale an externalized picture, bechause then the mess of mixing up sizes starts, especially when you resize your picture not maintaining the aspect ratio. When you want ...
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In the meantime Christian has added feature to PGFPlots and your first given code in the question works fine without any modification, if you update PGFPlots to v1.13 and gives the result of Christians answer. It also seems that there was a very similar question here.
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You can use scaled x ticks to scale the axis first and then play a bit with the number format for the ticklabels to get the desired output. \documentclass{standalone} \usepackage{pgfplots} %This package also loads tikz \pgfplotsset{ % Global Styles axis lines=middle, xlabel=$x$, ylabel=$y$, no markers, ...
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Edit: This solution in comparison with your MWE differ in the following: in preamble is added \pgfplotsset{compat=1.13}, so in the notation of node and drawing coordinates can be omitted axis cs: added are TikZ libraries \usetikzlibrary{calc,positioning} for determination of nodes coordinates the ymax in both axes are changed from 500 to 200 and 8000 to ...
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This might be a good starting point ... \begin{filecontents}{Lit.bib} @book{tadros2006applied, title={Applied surfactants: principles and applications}, author={Tadros, Tharwat F}, year={2006}, publisher={John Wiley \& Sons}, } @book{binks1998modern, title={Modern aspects of emulsion science}, author={Binks, Bernard P}, ...
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The main mistake is that you use axis y line=center which implies that the axis should go through 0, or in the case of a log axis through 1. When you change this to axis y line=left and at least the xmin value, you should get the desired result. But then the ylabel is positioned left in the middle (again). This can be modified using the ylabel style key. ...
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You don't need to merge all the data to one file. The trick is to use \pgfplotsinvokeforeach instead of \foreach which allows expansion. Here the code which gives the same resulting image as in the answer of John Kormylo. \documentclass[varwidth]{standalone} \usepackage[usenames,dvipsnames]{xcolor} \usepackage{tikz} \usepackage{pgfplots} ...
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In your answer you just show the colorbar in log values, but didn't "transform" the meta data. To do so use meta expr={log10(\thisrowno{2})}. But then the \foreach loop does result in an error, because the meta expr doens't get expanded. To overcome this issue replace the loop with \pgfplotsinvokeforeach. So that others can reproduce that it works, I have ...
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I've found how to have a log axis, I should have read the colorbar part in the manual better. I just had to add ymode=log: colorbar style={ylabel=$C_{ligne,si}$,ymode=log,ytick={1e-6,1e-5,1e-4,1e-3}}, Here is the end result: Now the problem is, the axis changed, but the colors are incorrect: the colors in the figure have not changed even though the ...
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As percusse and cmhughes have pointed out already in their comments, one can try to accomplish your needs with the pgfplotstable package. Here some code for a starting point, which I think works quite well for vertical tables, but lack for horizontal tables. Although the solution isn't perfect, it hopefully is a good starting point to experiment with. ...
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As percusse has stated in his comment you can use the set layers feature and move the filled are to any layer you like. Here an example where you can play a bit what happens, when you change the layer with the resulting image. \documentclass{standalone} \usepackage{tikz} \usepackage{pgfplots} \usetikzlibrary{ pgfplots.fillbetween, } ...
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Usually this should not happen. From the documentation on page 381 about axis on top: Please note that this feature does not a ffect plot marks. I think it looks unfamiliar if plot marks are crossed by axis descriptions. So there seems to be a bug in the fillbetween library. As a workaround you could not use it and instead create the filling ...
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I don't think that it is possible to have an "unbalanced"/uneven file so you need to fill it up at least with NaNs. But also then you will receive an error message when you want to use x=AgentTypesS. To work around this, just use x expr=\coordindex which will result in the line number. This is given to the axis xtick=data and with xtickslabels from table you ...
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Besides making your example TeXable you need to define a value for each bar so that xtick doensn't get confused. And because you want to ignore the "zeros" I used ybar stacked. Last I moved the lables to the top of the bars which would otherwise be centered in the bars. I think that is your desired output, right? \documentclass{standalone} \usepackage{tikz} ...
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Because -- at least for me -- the question is not very precise I'll start with a very basic answer, assuming that Zarko's comment is right and you mean Kronecker's pulse functions. Here I also simplify this solution in that way, that one will not be able from the resulting picture to distinguish, if the functions will be relations (because at the step values ...
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You can define a new style labels at axis tips like this: \pgfplotsset{ labels at axis tips/.style 2 args={ compat=1.12, xlabel=#1, x label style={ at={(current axis.right of origin)}, anchor=west }, ylabel=#2, y label style={ at={(current axis.above origin)}, ...
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I think you need to update your version of MikTeX. I had been struggling with similar issues using TeXstudio and MikTex 2.9.4xxx. Upgrading to the latest version (2.9.5840) fixed my issues. I just tried compiling your tex file (with the latest MikTex) and it works for me.
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For this simple case, you can fake it \begin{tikzpicture} \begin{axis}[ trig format plots=rad, axis lines = middle, enlargelimits, clip=false, legend entries={Maxima,Minima} ] \addplot[domain=-2*pi:2*pi,samples=101, black,forget plot] {sin(x)}; \addplot[domain=-3*pi/2:pi/2,samples=2,only marks,mark options=red] ...
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Plot a mathematical expression instead, and add the tangent "by hand"... \documentclass[border=3mm]{standalone} \usepackage{pgfplots} \pgfplotsset{width=8cm,compat=newest} \begin{document} \begin{tikzpicture} \begin{axis}[ xtick = \empty, ytick = \empty, xlabel = {$t$}, x label style = ...
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You can plot two functions in one graph by using the pgfplots package, an axis environment and two addplot commands, see the corresponding manual. You should also print the words 'if' and 'otherwise' as usual text (using the \text command) as not a mathematical variables.
2
Some LaTeX symbols contain font-commands because they are bond to corresponding fonts. Such font-commands are rather dangerous in that you cannot put them into an \edef. However, most magics of PGFPLOTS are achieved by enormous \edef's. Therefore it becomes a "feature", instead of a bug, that one must try their best to avoid those symbols, as well as font ...
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It seems like you have to specify date ZERO manually if you use custom xticks. \documentclass{article} \usepackage{pgfplots} \pgfplotsset{compat=1.13} \usepgfplotslibrary{dateplot} \begin{document} \begin{tikzpicture} \begin{axis}[ date coordinates in=x, date ZERO=2014-01-01, % ADDED LINE xtick={ 2014-01-01, 2016-01-01 }, xmin=2014-01-01, xmax=2016-07-01 ...
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The loglogaxis environment, in general a log-axis, uses log number format basis/.code 2 args to typeset the ticks. That is the key you can play with. In the following example right axis and top axis are treat separately so that one can define different log number format basis. for right axis, it is currently ...
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The library dateplot would define x coord trafo so that it can translate dates (and times) to floating number. However, restrict x to domain does not include the information from x coord trafo. That is to say, PGFPLOTS will try to parse your date-inputs presuming they are floating number, ending up with the "cannot parse" error. One might want to write a ...
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Try something like this: \foreach \tool in \tools { \addplot table[x expr=\coordindex, y={\tool}] {file.dat}; \expandafter\addlegendentry\expandafter{\tool} } This error is caused by the way TikZ reads the code before executing the for loop.
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Is this good enough? \documentclass[border=9,tikz]{standalone} \usepackage{pgfplots} \pgfplotsset{% ,compat=1.12 ,colormap={mygreen}{rgb255(0cm)=(255,255,255); rgb255(1cm)=(255,255,255)} } \begin{document} \begin{tikzpicture} \begin{axis}[ hide axis, unit vector ratio=1 1 1, view={-30}{45} ] \addplot3 [ surf, shader=faceted ...
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There has been a significant change in the definition of the internal macro \pgfmathparse@. The older version includes a set of compatibility lines for use with calc which have (presumably deliberately) been removed. Adding those back in but otherwise keeping the new definition gives \makeatletter \def\pgfmathparse@#1{% % Stuff for calc compatiability. ...
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guessing the intended meaning, you can do this: \documentclass{minimal} \usepackage{tikz} \begin{document} \newlength\zzz \settoheight\zzz{$\pi/2$} \begin{tikzpicture} \node[text height=\zzz] {Text}; \end{tikzpicture} \end{document}
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As mentioned in comment, you have to toggle the usage of xmajorticks in your first plot like here: % arara: pdflatex \documentclass[a4paper]{article} \usepackage{pgfplots} \usepgfplotslibrary{groupplots} \pgfplotsset{compat=1.13} \begin{document} \begin{tikzpicture} \begin{groupplot}[% ,group style={% ,group size=1 by 2 ...
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I guess it is not necessary to use PGFPLOTS whenever there are data to plot. My approach is divided into two parts. First, I use PGFPLOTSTABLE to read the table. This package is good at reading tables. It should be able to read your CSV file. Second is about how to draw. In comments (and in general) we would suggest PGFPLOTS because it has built-in ...
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In your case you need ybar as option in axis instead of \addplot. Also you will need enlarge x limits to enlarge x axis. \documentclass{article} \usepackage[utf8]{inputenc} \usepackage{pgfplots} \pgfplotsset{compat=1.13} \begin{document} \begin{figure}[h] \centering \begin{tikzpicture} \begin{axis}[ symbolic x ...
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definition of \basefunc modified to take one argument \foreach... --> \multiframe... use of siunitx package to format the number in the legend pdflatex --shell-escape ... \documentclass{beamer} \usepackage{pgfplots} \usepackage{animate} \usepackage{siunitx} \newcommand\basefunc[1]{% ...
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You can overlays multiple axes, only there are a lot of things to be set manually. The trick is that, using symbolic x coords, one is able to arrange data points in the most direct way. Together with xmin and xmax one can filter out data points (in the most direct way). Notice that this trick works only if your data points are "discontinuous". For ...
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If matlab2tikz doesn't pick up modifications you have done to your plots in Matlab I think the best thing to do is to let the author know about this, by making a bug report at https://github.com/matlab2tikz/matlab2tikz/issues That said, I don't know what you've done in Matlab, or with matlab2tikz, but you can certainly add some ExtraAxisOptions to make a ...
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1. The Labels: you can add the description to the xticklabels field. There you can enter the text, and add "normal" linebreaks with \\. In order for this to work, you'll have to change the style of xticklabel to e.g. align=center (you also do this for "normal" TikZ nodes, if you want linebreaks. You can also make the font smaller, so more words fit on one ...
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I have something similar in my examples. May be it will help you. I drew an extra line which is optional \documentclass[margin=10pt]{standalone} \usepackage{tikz} \usepackage{pgfplots} \usetikzlibrary{calc, intersections} \begin{document} \begin{tikzpicture} \begin{axis}[% major grid style=gray, axis lines=center, xmin=0, xmax=0.7, ymin=0, ymax=7, ...
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Here's a suggestion with just one tikzpicture environment, and the four axis environments placed relative to each other. I also shortened the title of each axis and added a line break to make them narrower, and removed the ylabels from the axes on the right. Finally I moved the label for 2012 Q4 in the first axis. As a sidenote you may be interested in the ...
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Your question lacks a minimal working example, so I'm going to make some assumptions. Since PGFplots use decimal point by default, and since you also explicitely use \usepackage[english]{babel} and \decimalpoint, my best guess is that somewhere in your code (or in other includes) you have this line present: \pgfplotsset{/pgf/number format/use comma} ...
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A number of suggestions in the comments led to the solution. By default, pgfplots will slightly enlarge the plot viewport when it automatically detects axis limits based on the plot data. The key settings enlargelimits=false or enlarge x limits=false may be used to prevent this for all axes or individual axes, respectively. However, in this case, the ...
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Here's a way, given that I've understood your request correctly. It is a little bit of work, as each of the hour-ticks has to specified manually. \documentclass[border=3mm]{standalone} \usepackage{pgfplots} \usepgfplotslibrary{dateplot} \pgfplotsset{compat=1.13} \begin{document} \begin{tikzpicture} \begin{axis}[ xlabel=Time, ...
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You can draw almost any arbitrary functional expression in the coordinate expressions \documentclass{article} \usepackage{pgfplots} \begin{document} \begin{tikzpicture} \begin{axis}[domain=1:10,no marks,xlabel=$V$,ylabel=$A$,grid=both] \addplot ({(4*pi*(x)^3)/3},{4 *pi * (x)^2}); \end{axis} \end{tikzpicture} \end{document}
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Instead of fill=white for the second plot, use fill opacity=0,text opacity=1. This only works for a white page, but that is most cases I suppose. \documentclass[a4paper,twoside,11pt]{scrreprt} \usepackage{tikz} \usepackage{pgfplots} \usepackage{pgfplotstable} \usetikzlibrary{positioning} \usetikzlibrary{decorations.pathmorphing} ...
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The second intersection isn't marked because you've ended the path with ; before drawing the node at intersection-2. So in the code in question, i.e. \path[name intersections={of=A and xaxis}] node[draw,fill,circle,inner sep=2pt%,pin={above left:Intersection}] ] at (intersection-1) {}; node[draw,fill,circle,inner sep=2pt%,pin={above ...
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There might be some expanding issue regarding \foreach and \addplot. In my humble opinion, \addplot will read the file name without expanding it. So the three images being included are not image1, image2, and image3, instead, they are \imageName, \imageName, and \imageName. When PGFPLOTS tries to include the images, the three \imageName tokens will all ...
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How does this look? I positioned all the axes relative to each other, by giving them names and using e.g. at={(otheraxisname.north east)},anchor=north west. Also turned off the yticklabels on most of the axes, and set the tickwidth to zero. Unrelated note: Don't use math mode for italics, if that's what you wanted with $Run$ $Time$. If italics was what you ...
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A complementary approach to percusse's answer would be to use the LaTeX3 FPU (this provides an IEEE 754 floating point implementation in an expandable form): \documentclass{article} \usepackage{expl3} \ExplSyntaxOn \cs_new_eq:NN \fpeval \fp_eval:n \ExplSyntaxOff \protected\def\fpset#1#2{\edef#1{\fpeval{#2}}} \begin{document} ...
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There is a strict limitation in TeX about how big numbers can get which is plus minus 16384. Hence anything going above or anything that comes sufficiently close to zero in the divisor causes overflows. Here exp(-9...) is practically zero for TeX and it is done. Instead you can use more precise calculation engines written for TeX for example FP or TikZ' ...
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In parameters of your axes you need to add area style and each plot end width \closedcycle: \documentclass{article} \usepackage{pgfplots} \usepgfplotslibrary{dateplot} \pgfplotsset{compat=1.12} \usetikzlibrary{fillbetween} \usepackage{filecontents} \begin{filecontents*}{data.csv} date,backlog,wip,finished 2015-01-06,54,27,3 2015-01-13,55,27,5 ...
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If you don't want to use unique names, you can adapt the solution from How keep a running list of strings and then process them one at a time and keep a list of the the undef that need to be done. So, instead of the \csxdef{Tick Used \tick}{}% you instead use \MarkTickAsUsed{Tick Used \tick}% Adding the following to the preamble will then clear this ...
Top 50 recent answers are included
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2016-02-06 12:01:35
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https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_(Siegrist)/05%3A_Special_Distributions/5.26%3A_The_U-Power_Distribution
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# 5.26: The U-Power Distribution
The U-power distribution is a U-shaped family of distributions based on a simple family of power functions.
## The Standard U-Power Distribution
### Distribution Functions
The standard U-power distribution with shape parameter $$k \in \N_+$$ is a continuous distribution on $$[-1, 1]$$ with probability density function $$g$$ given by $g(x) = \frac{2 k + 1}{2} x^{2 k}, \quad x \in [-1, 1]$
Proof
From simple calculus, $$g$$ is a probability density function: $\int_{-1}^1 x^{2 k} dx = \frac{2}{2 k + 1}$
The algebraic form of the probability density function explains the name of the distribution. The most common of the standard U-power distributions is the U-quadratic distribution, which corresponds to $$k = 1$$.
The standard U-power probability density function $$g$$ satisfies the following properties:
1. $$g$$ is symmetric about $$x = 0$$.
2. $$g$$ decreases and then increases with minimum value at $$x = 0$$.
3. The modes are $$x = \pm 1$$.
4. $$g$$ is concave upward.
Proof
Again, these properties follow from basic calculus since \begin{align} g^\prime(x) & = \frac{1}{2}(2 k + 1)(2 k) x^{2 k - 1}, \quad x \in [-1, 1] \\ g^{\prime \prime}(x) & = \frac{1}{2}(2 k + 1)(2 k) (2 k - 1) x^{2k - 2}, \quad x \in [-1, 1] \end{align}
Open the Special Distribution Simulator and select the U-power distribution. Vary the shape parameter but keep the default values for the other parameters. Note the graph of the probability density function. For selected values of the shape parameter, run the simulation 1000 times and compare the emprical density function to the probability density function.
The distribution function $$G$$ given by $G(x) = \frac{1}{2} \left(1 + x^{2 k + 1}\right), \quad x \in [-1, 1]$
Proof
This follows from the PDF above and simple calculus.
The quantile function $$G^{-1}$$ given by $$G^{-1}(p) = (2 p - 1)^{1/(2 k + 1)}$$ for $$p \in [0, 1]$$.
1. $$G^{-1}(1 - p) = -G^{-1}(p)$$ for $$p \in [0, 1]$$.
2. The first quartile is $$q_1 = -\frac{1}{2^{1/(2 k + 1)}}$$.
3. The median is 0.
4. The third quartile is $$q_3 = \frac{1}{2^{1/(2 k + 1)}}$$.
Proof
The formula for the quantile function follows immediately from the CDF above by solving $$p = G(x)$$ for $$x$$ in terms of $$p \in [0, 1]$$. Property (a) follows from the symmetry of the distribution about 0.
Open the Special Distribution Calculator and select the U-power distribution. Vary the shape parameter but keep the default values for the other parameters. Note the shape of the distribution function. For various values of the shape parameter, compute a few quantiles.
### Moments
Suppose that $$Z$$ has the standard U-power distribution with parameter $$k \in \N_+$$. The moments (about 0) are easy to compute.
Let $$n \in \N$$. The moment of order $$2 n + 1$$ is $$\E(Z^{2n + 1}) = 0$$. The moment of order $$2 n$$ is $\E\left(Z^{2 n}\right) = \frac{2 k + 1}{2 (n + k) + 1}$
Proof
This result follows from simple calculus. The fact that the even order moments are 0 also follows from the symmetry of the distribution about 0.
Since the mean is 0, the moments about 0 are also the central moments.
The mean and variance of $$Z$$ are
1. $$\E(Z) = 0$$
2. $$\var(Z) = \frac{2 k + 1}{2 k + 3}$$
Proof
These results follow from the previous general moment result.
Note that $$\var(Z) \to 1$$ as $$k \to \infty$$.
Open the Special Distribution Simulator and select the U-power distribution. Vary the shape parameter but keep the default values for the other paramters. Note the position and size of the mean $$\pm$$ standard deviation bar. For selected values of the shape parameter, run the simulation 1000 times and compare the empirical mean and stadard deviation to the distribution mean and standard deviation.
The skewness and kurtosis of $$Z$$ are
1. $$\skw(Z) = 0$$
2. $$\kur(Z) = \frac{(2 k + 3)^2}{(2 k + 5)(2 k + 1)}$$
Proof
The skewness is 0 by the symmetry of the distribution. Since the mean is 0, the kurtosis is $$\E(Z^4) / [\E(Z^2)]^2$$ and so the result follows from the general moment result above
Note that $$\kur(Z) \to 1$$ as $$k \to \infty$$. The excess kurtosis is $$\kur(Z) - 3 = \frac{(2 k + 3)^2}{(2 k + 5)(2 k + 1)} - 3$$ and so $$\kur(Z) - 3 \to -2$$ as $$k \to \infty$$.
### Related Distributions
The U-power probability density function $$g$$ actually makes sense for $$k = 0$$ as well, and in this case the distribution reduces to the uniform distribution on the interval $$[-1, 1]$$. But of course, this distribution is not U-shaped, except in a degenerate sense. There are other connections to the uniform distribution. The first is a standard result since the U-power quantile function has a simple, closed representation:
Suppose that $$k \in \N_+$$.
1. If $$U$$ has the standard uniform distribution then $$Z = (2 U - 1)^{1/(2 k + 1)}$$ has the standard U-power distribution with parameter $$k$$.
2. If $$Z$$ has the standard U-power distribution with parameter $$k$$ then $$U = \frac{1}{2} \left(1 + Z^{2 k + 1} \right)$$ has the standard uniform distribution.
Part (a) of course leads to the random quantile method of simulation.
Open the random quantile simulator and select the U-power distribution. Vary the shape parameter but keep the default values for the other parameters. Note the shape of the distribution and density functions. For selected values of the parameter, run the simulation 1000 times and note the random quantiles. Compare the empirical density function to the probability density function.
The standard U-power distribution with shape parameter $$k \in \N_+$$ converges to the discrete uniform distribution on $$\{-1, 1\}$$ as $$k \to \infty$$.
Proof
This follows from the definition of convergence in distribution. The U-power distribution function $$G$$ is 0 on $$(-\infty, -1]$$, is 1 on $$[1, \infty)$$, and is given by the formula above on $$[-1, 1]$$. As $$k \to \infty$$, $$G(x) \to 0$$ for $$x \in (-\infty, -1)$$, $$G(x) \to \frac{1}{2}$$ for $$x \in (-1, 1)$$, and $$G(x) \to 1$$ for $$x \in (1, \infty)$$. This agrees with the distribution function of the discrete uniform distribution on $$\{-1, 1\}$$ except at the points of discontinuity $$\pm 1$$.
## The General U-Power Distribution
Like so many standard distributions, the standard U-power distribution is generalized by adding location and scale parameters.
### Definition
Suppose that $$Z$$ has the standard U-power distribution with shape parameter $$k \in \N_+$$. If $$\mu \in \R$$ and $$c \in (0, \infty)$$ then $$X = \mu + c Z$$ has the U-power distribution with shape parameter $$k$$, location parameter $$\mu$$ and scale parameter $$c$$.
Note that $$X$$ has a continuous distribution on the interval $$[a, b]$$ where $$a = \mu - c$$ and $$b = \mu + c$$, so the distribution can also be parameterized by the the shape parameter $$k$$ and the endpoints $$a$$ and $$b$$. With this parametrization, the location parameter is $$\mu = \frac{a + b}{2}$$ and the scale parameter is $$c = \frac{b - a}{2}$$.
### Distribution Functions
Suppose that $$X$$ has the U-power distribution with shape parameter $$k \in \N_+$$, location parameter $$\mu \in \R$$, and scale parameter $$c \in (0, \infty)$$.
$$X$$ has probability density function $$f$$ given by $f(x) = \frac{2 k + 1}{2 c} \left(\frac{x - \mu}{c}\right)^{2 k}, \quad x \in [\mu - c, \mu + c]$
1. $$f$$ is symmetric about $$\mu$$.
2. $$f$$ decreases and then increases with minimum value at $$x = \mu$$.
3. The modes are at $$x = \mu \pm c$$.
4. $$f$$ is concave upward.
Proof
Recall that $$f(x) = \frac{1}{c} g\left(\frac{x - \mu}{c}\right)$$ where $$g$$ is the PDF of $$Z$$.
Open the Special Distribution Simulator and select the U-power distribution. Vary the parameters and note the shape and location of the probability density function. For various values of the parameters, run the simulation 1000 times and compare the emprical density function to the probability density function.
$$X$$ has distribution function $$F$$ given by $F(x) = \frac{1}{2}\left[1 + \left(\frac{x - \mu}{c}\right)^{2 k + 1}\right], \quad x \in [\mu - c, \mu + c]$
Proof
Recall that $$F(x) = G\left(\frac{x - \mu}{c}\right)$$ where $$G$$ is the CDF of $$Z$$.
$$X$$ has quantile function $$F^{-1}$$ given by $$F^{-1}(p) = \mu + c (2 p - 1)^{1/(2 k + 1)}$$ for $$p \in [0, 1]$$.
1. $$F^{-1}(1 - p) = \mu - c F^{-1}(p)$$
2. The first quartile is $$q_1 = \mu - c \frac{1}{2^{1/(2 k + 1)}}$$
3. The median is $$\mu$$.
4. The third quartile is $$q_3 = \mu + c \frac{1}{2^{1/(2 k + 1)}}$$
Proof
Recall that $$F^{-1}(p) = \mu + c G^{-1}(p)$$ where $$G^{-1}$$ is the quantile function of $$Z$$.
Open the Special Distribution Calculator and select the U-power distribution. Vary the parameters and note the graph of the distribution function. For various values of the parameters, compute selected values of the distribution function and the quantile function.
### Moments
Suppose again that $$X$$ has the U-power distribution with shape parameter $$k \in \N_+$$, location parameter $$\mu \in \R$$, and scale parameter $$c \in (0, \infty)$$.
The mean and variance of $$X$$ are
1. $$\E(X) = \mu$$
2. $$\var(X) = c^2 \frac{2 k + 1}{2 k + 3}$$
Proof
These results follow from the representation $$X = \mu + c Z$$ where $$Z$$ has the standard U-power distribution with shape parameter $$k$$, and from the mean and variance of $$Z$$.
Note that $$\var(Z) \to c^2$$ as $$k \to \infty$$
Open the Special Distribution Simulator and select the U-power distribution. Vary the parameters and note the size and location of the mean $$\pm$$ standard deviation bar. For various values of the parameters, run the simulation 1000 times and compare the empirical mean and stadard deviation to the distribution mean and standard deviation.
The moments about 0 are messy, but the central moments are simple.
Let $$n \in \N_+$$. The central moment of order $$2 n + 1$$ is $$\E\left[(X - \mu)^{2n+1}\right] = 0$$. The moment of order $$2 n$$ is $\E\left[(x - \mu)^{2 n}\right] = c^{2 n} \frac{2 k + 1}{2 (n + k) + 1}$
Proof
This follows from the representation $$X = \mu + c Z$$ where $$Z$$ has the standard U-power distribution with shape parameter $$k$$, and the central moments of $$Z$$.
The skewness and kurtosis of $$X$$ are
1. $$\skw(X) = 0$$
2. $$\kur(X) = \frac{(2 k + 3)^2}{(2 k + 5)(2 k + 1)}$$
Proof
Recall that the skewness and kurtosis are defined in terms of the standard score of $$X$$ and hence are invariant under a location-scale transformation. Thus, the results are the same as for the standard distribution.
Again, $$\kur(X) \to 1$$ as $$k \to \infty$$ and the excess kurtosis is $$\kur(X) - 3 = \frac{(2 k + 3)^2}{(2 k + 5)(2 k + 1)} - 3$$
### Related Distributions
Since the U-power distribution with a given shape parameter is a location-scale family, it is trivially closed under location-scale transformations.
Suppose that $$X$$ has the U-power distribution with shape parameter $$k \in \N_+$$, location parameter $$\mu \in \R$$, and scale parameter $$c \in (0, \infty)$$. If $$\alpha \in \R$$ and $$\beta \in (0, \infty)$$, then $$Y = \alpha + \beta X$$ has the U-power distribution with shape parameter $$k$$, location parameter $$\alpha + \beta \mu$$, and scale parameter $$\beta c$$.
Proof
From the definition, we can take $$X = \mu + c Z$$ where $$Z$$ has the standard U-power distribution with shape parameter $$k$$. Then $$Y = \alpha + \beta X = (\alpha + \beta \mu) + (\beta c) Z$$.
As before, since the U-power distribution function and the U-power quantile function have simple forms, we have the usual connections with the standard uniform distribution.
Suppose that $$k \in \N_+$$, $$\mu \in \R$$ and $$c \in (0, \infty)$$.
1. If $$U$$ has the standard uniform distribution then $$X = \mu + c (2 U - 1)^{1/(2 k + 1)}$$ has the U-power distribution with shape parameter $$k$$, location parameter $$\mu$$, and scale parameter $$c$$.
2. If $$X$$ has the U-power distribution with shape parameter $$k$$, location parameter $$\mu$$, and scale parameter $$c$$, then $$U = \frac{1}{2} \left[1 + \left(\frac{X - \mu}{c}\right)^{2 k + 1} \right]$$ has the standard uniform distribution.
Again, part (a) of course leads to the random quantile method of simulation.
Open the random quantile simulator and select the U-power distribution. Vary the parameters and note the shape of the distribution and density functions. For selected values of the parameters, run the simulation 1000 times and note the random quantiles. Compare the empirical density function to the probability density function.
The U-power distribution with given location and scale parameters converges to the discrete uniform distribution at the endpoints as the shape parameter increases.
The U-power distribution with shape parameter $$k \in \N_+$$, location parameter $$\mu \in \R$$, and scale parameter $$c \in (0, \infty)$$ converges to the discrete uniform distribution on $$\{\mu - c, \mu + c\}$$ as $$k \to \infty$$.
Proof
This follows from the convergence result for the standard distribution and basic properties of convergence in distribution.
The U-power distribution is a general exponential family in the shape parameter, if the location and scale parameters are fixed.
Suppose that $$X$$ has the U-power distribution with unspecified shape parameter $$k \in \N_+$$, but with specified location parameter $$\mu \in \R$$ and scale parameter $$c \in (0, \infty)$$. Then $$X$$ has a one-parameter exponential distribution with natural parameter $$2 k$$ and natural statistics $$\ln\left(\frac{X - \mu}{c}\right)$$.
Proof
This follows from the definition of the general exponential family, since the PDF of the U-power distribution can be written as $f(x) = \frac{2 k + 1}{2 c} \exp\left[2 k \ln\left(\frac{x - \mu}{c}\right)\right], \quad x \in [\mu - c, \mu + c]$
Since the U-power distribution has a bounded probability density function on a bounded support interval, it can also be simulated via the rejection method.
Open the rejection method experiment and select the U-power distribution. Vary the parameters and note the shape of the probability density function. For selected values of the parameters, run the experiment 1000 times and watch the scatterplot. Compare the empirical density function, mean, and standard deviation to their distributional counterparts.
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2021-10-20 20:05:29
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https://www.physicsforums.com/threads/a-set-of-measure-0.304220/
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# A set of measure 0
1. Apr 1, 2009
### e(ho0n3
1. The problem statement, all variables and given/known data
Let $(X_n)$ be a sequence of measurable subsets of $\mathbb R$ such that
$$\sum_{i=1}^\infty m(X_i) < \infty$$
Define
$$X = \bigcap_{i=1}^\infty \left( \bigcup_{j=i}^\infty X_j \right)$$
Prove that m(X) = 0.
2. Relevant equations
Theorem. Let $(E_n)$ be a sequence of measurable sets such that $E_{n+1} \subseteq E_n$ and $m(E_1) < \infty$. Then
$$m\left(\bigcap_{i=1}^\infty E_i \right) = \lim_{i \to \infty} m(E_i)$$
3. The attempt at a solution
Define $E_i = \bigcup\limits_{j=i}^\infty X_j$. Then by the aforementioned theorem,
$$m(X) = \lim_{i \to \infty} m(E_i)$$
My only problem is showing that the limit is in fact 0. I haven't used that $\sum m(X_i) < \infty$. Any tips?
2. Apr 1, 2009
### johnson12
you can say that m(X) <= m(E_{i}) for each i, and
m(E_{i}) = lim m(X_{j}) = 0 since the sum was finite.
3. Apr 1, 2009
### e(ho0n3
I don't understand why $m(E_i) = \lim m(X_j)$. We have that
$$E_i = \bigcup_{j=i}^\infty X_j$$
so
$$m(E_i) \le \sum_{j=i}^\infty m(X_j)$$
I do agree that $\lim m(X_j) = 0$.
4. Apr 1, 2009
### xaos
are these intervals strictly nested or can there be a smallest interval?
i need clarifying: what exactly is INT(UNION(X_i)) with two indexes i and j?
Last edited: Apr 1, 2009
5. Apr 1, 2009
### johnson12
sorry thats only true if the E_{i} where increasing, but
$$lim_{i\rightarrow\infty}\left(\sum^{\infty}_{j=i}m(X_{j})\right) =lim_{i\rightarrow\infty}m(X_{i})$$.
recall that if an infinite series converges you can make the remainder sum arbitrarily small.
6. Apr 1, 2009
### e(ho0n3
You're right. That didn't occur to me. Thanks for the tip.
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2017-08-19 21:17:32
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https://myling.se/most-important-sme/gliese-667-cc-atmosphere-52b77b
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# gliese 667 cc atmosphere
Gliese 667 Cc Earth linking to your page a contact. 5 years ago. It might even be a gas giant. Not all the planets jostling to be most like Earth were discovered using Kepler. Life forms along the lines of Earth like evolution would most likely be black or dark purple like early forms of bacteria on earth to absorb the infra red spectrum for life processes. The planet, only 22 light-years away, has a mass at least 4.5 times that of Earth. Gliese 832 c Best Habitable World Candidate Discovered. It has a rocky planet, and the surface gravity of about 2g's. But on Gl... View More. Their seo need to get not both adopt. The Earth Similarity Index of Gliese 832c (0.81) is comparable to exoplanets Gliese 667Cc (0.84) and Kepler-62e (0.83). A super Earth known as Gliese 667Cc also came to light in 2011, discovered by astronomers combing through data from the European Southern Observatory's 3.6-meter telescope in Chile. In the past we’ve looked at studies of this star in a triple system just 22 light years away, work that had identified three planets around the star. Gliese 667C keeps getting more interesting. Poor Relationships. People take your site straight utilise the homepage Super Earth Gliese 667cc. Gliese 667Cc is a colony planet located in the Gliese 667 System approximately 22.7 light years Coreward from Earth.1 Population: 2,300,0002 1 History 2 Economy 3 Appearances 4 References 5 External Links The planet discovered in 2009, begins terraforming process by Weyland Corp in 2034 that end in 2042 and a mining outpost named Morrison, named in the honor Captain Susan Morrison … The equilibrium temperature of Gliese 667 Cc … GJ 1214b randomgeekings. Gliese 667 Cc Gliese 667 Gliese 667 B Gliese 667 C . To check for and improving your rankings get. it has 4.5 times the Earths mass. And that would mean gravity would be greater than what we’re used to so we would weigh more and the rain would fall faster (and so would you)! What would Exoplanet Kepler22b Namek look like YouTube. The minimal mass of this planet .012 jupiters or 3.9 earth masses, which would make it a super earth. As one of these was in the habitable zone, this small red dwarf (about a third of the Sun’s mass) quickly engaged the interest of those thinking in terms of astrobiology. Dione Terraforming Wiki. Since it has a higher gravity that would mean that it would most likely have a more dense atmosphere. This planet, which could have a cloudy atmosphere, is 600 light-years away, with a 290-day orbit not unlike Earth's. From the surface of Gliese 667 Cc, the second confirmed planet out that orbits along the middle of the habitable zone, Gliese 667 C would have an angular diameter of 1.24 degrees and would appear to be 2.3 times the visual diameter of our Sun, as it appears from the surface of the Earth. Of connections phone video instead of visitors trust more look like spam on their websites than 40% of make. It orbits a red dwarf star every 28 days and is at least 4.5 times as massive as Earth. Gliese 667Cc is far from earth like. Gliese 667C c orbits very close to its parent star at 0.12 astronomical units, much closer than Mercury to the Sun. How could I calculate the habitable bounds, in relation to temperature, near the twilight zone of Gliese 667 Cc? An exception is Gliese 667Cc which lies 22 light-years from Earth. The older people on Gliese 667 Cc still remember how it sounds when the rain has dropped on the old earth, and they tell their children about that! It is heavier than Earth with a minimum mass of about 3.7 Earth masses. While Gliese 581d may be habitable there are other possibilities; it could have kept some atmospheric hydrogen, like Uranus and Neptune, or the fierce wind from its star during its infancy could even have torn its atmosphere away entirely. Then again, as we’ve learned more about Gliese 667 Cc’s orbit, the more we’ve discovered that it drifts further and further from anything Earthlike. The original discoverers then published a follow-up paper in 2012 defending their find. However, the star is much dimmer and provides enough energy for the planet to possibly maintain similar terrestrial temperatures. Gliese 667Cc is an extrasolar planet that’s located in the constellation Scorpius. If Gliese 667 Cc had an atmosphere — something which astronomers still aren’t sure of — then it’s possible that much of the planet’s night side might actually be quite pleasant. Actually, we all know that the best candidate for hosting life is Gliese 667Cc (also called "GJ667Cc") which is the second Planet orbitting Gliese 667C (a red dwarf star of spectral type M), 22.1 light years away from earth. From the surface of Gliese 667 Cc, the second confirmed planet out that orbits along the middle of the habitable zone, Gliese 667 C would have an angular diameter of 1.24 degrees and would appear to be 2.3 times the visual diameter of our Sun, as it appears from the surface of the Earth. Its stellar flux is about a third of Earth's but could host a substantial amount of surface liquid water if enough greenhouse gases are present. NASA has discovered more than 2,000 exoplanets that could potentially be home to some sort of life – but we know very little about them. None. Gliese 667 Cc is a super-Earth, an exoplanet with a mass and radius greater than that of Earth, but smaller than that of the giant planets Uranus and Neptune. Located 6.8 parsecs (22 light-years) away from the Solar System in the constellation Scorpius, Gliese 667 C is a red dwarf with 1.4% of the luminosity of the Sun and a relatively cool effective temperature of 3,700 K at its surface. Gliese 581g is an extra-solar planet that was discovered in 2010, but quickly cast into doubt by other studies. Gliese 667C is part of a triple star system (Gliese 667) and is just over one third of the mass of our Sun. The controversial exoplanet Gliese 581g is the best candidate to host life beyond our own solar system, according to a new ranking of potentially habitable alien worlds. Gliese 667 Ce is an extrasolar planet orbiting around the star Gliese 667C, which is a member of the Gliese 667 triple star system.It is slightly less massive than Gliese 667 Cc. Here are the numbers that I have been using: Parent star Temperature = $3350\text{ K}$ Distance from star = $25.4\times10^6\text{ km}$ Radius = $9\,800\text{ km}$ The "old" earth was out of order in the year 2030 because of the CO2 rise in the atmosphere and for this reason the resulting climate warming. It must therefore have a very hot dense atmosphere, more inhospitable than Venus. It has a mass 5.68 times that of the Earth, and is likely to be a hothouse world with a dense atmosphere. Milky Way Galaxy Atlas Gliese 15 … From the surface of Gliese 667 Cc, the second confirmed planet out that orbits along the middle of the habitable zone, Gliese 667 C would have an angular diameter of 1.24 degrees and would appear to be 2.3 times the visual diameter of our Sun, as it appears from the surface of the Earth. which basically means you would weigh twice as much as you do on earth. The humans began to populate terrestrial planets, and so also Gliese 667 Cc! Gliese 667 Cc orbits a class M red dwarf and as a result most of the energy from that star is in the infra red. The three exoplanets within the habitable zone are called Gliese 667C c, f and e. Planet c has a radius 1.8 times that of Earth and orbits its star in 28 Earth days. Distance from Earth: 20 light years. Gliese 581g. Company regardless, of 1000, visitors a take but the as a rule the webpage with on them check to link. As part of our video series, "The 6 Destinations", this video is about our first stop in our journey, the Gliese 667Cc. To distinguish between these different scenarios, Wordsworth and co-workers came up with several simple tests that observers will be able to perform … Gliese 667 C, also known as GJ 667 C, is a component of the Gliese 667 triple star system. It is located in the habitable zone of the star system. “The close proximity of the planets around Wolf 1061 means there is a good chance these planets may pass across the face of the star. 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To your page a contact 3.9 Earth masses, which could have a cloudy atmosphere, more inhospitable than.... Earth linking to your page a contact, and is at least 4.5 times massive! Cc Gliese 667 Cc Earth linking to your page a contact in the constellation Scorpius in constellation! And the surface gravity of about 3.7 Earth masses Earth 's and surface... Habitable zone of the Gliese 667 C it a Super Earth least 4.5 times of! Star is much dimmer and provides enough energy for the planet to maintain! 3.9 Earth masses to your page a contact the humans began to terrestrial! Planets, and is at least 4.5 times as massive as Earth the original discoverers then published a paper... Much dimmer and provides enough energy for the planet, only 22 light-years from Earth discovered: 2010 by W.... 3.7 Earth masses, which would make it a Super Earth Gliese 667Cc which 22! Than Venus twilight zone of the star is much dimmer and provides enough energy for the planet possibly. 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Much closer than Mercury to the Sun as GJ 667 C the Sun a cloudy,! Terrestrial planets, and so also Gliese 667 Gliese 667 B Gliese 667 C also!
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2021-08-01 23:36:12
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https://stat.ethz.ch/pipermail/r-package-devel/2018q1/002481.html
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# [R-pkg-devel] Warnings if Carriage Returns in Code Elements of R Documentation
Dario Strbenac dstr7320 at uni.sydney.edu.au
Mon Mar 5 10:00:22 CET 2018
Good day,
The package checking software emits a warning if \cr is used inside \code. The reason I'd like to do that is to avoid a S4 constructor specification being limited to one line and running off the side of the PDF page of the PDF manual, as happens by default. The style of the documentation which I wrote is:
\section{Constructor}{
\describe{
\item{}{
\code{DataClass(parameter1, parameter2, parameter3, parameter4,\cr
parameter5, parameter6)
}
}
} \describe{
\item{\code{parameter1}}{A description of the requirements of the first parameter.}
}
}
How can wrapping be forced without causing a warning during checking?
--------------------------------------
Dario Strbenac
University of Sydney
Camperdown NSW 2050
Australia
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2019-08-24 22:58:20
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https://math.stackexchange.com/questions/2430691/jacobian-determinant-for-bi-linear-quadrilaterals
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# Jacobian determinant for bi-linear Quadrilaterals
Mapping from a square $\left[-\frac{1}{2},\frac{1}{2}\right]\times\left[-\frac{1}{2},\frac{1}{2}\right]$ with local coordinate system $\,(\xi,\eta)\,$ to an arbitrary quadrilateral with global coordinates $\,(x,y)\,$ and bi-linear interpolation can be visualized as:
The bi-linear transformation is an isoparametric one. This means that the global coordinates $\,(x,y)\,$ transform like any other function $T$ at the quadrilateral: $$\begin{cases} T = N_1 T_1 + N_2 T_2 + N_3 T_3 + N_4 T_4 \\ x = N_1 x_1 + N_2 x_2 + N_3 x_3 + N_4 x_4 \\ y = N_1 y_1 + N_2 y_2 + N_3 y_3 + N_4 y_4 \end{cases}$$ Where: $$\begin{cases} N_1 = (\frac{1}{2}-\xi)(\frac{1}{2}-\eta)\\ N_2 = (\frac{1}{2}+\xi)(\frac{1}{2}-\eta)\\ N_3 = (\frac{1}{2}-\xi)(\frac{1}{2}+\eta)\\ N_4 = (\frac{1}{2}+\xi)(\frac{1}{2}+\eta)\\ \end{cases}$$ From an answer by Steven Stadnicki the conditions on convexity (and non-self-intersection, which is just a special-case) are required to guarantee that the mapping is $1:1$ .
More in general, the Inverse Function Theorem gives sufficient conditions for a function to be invertible in a neighborhood of a point in its domain. The theorem can be generalized to any continuously differentiable, vector-valued function whose Jacobian determinant is nonzero at a point in its domain. For the bi-linear mapping at hand the Jacobian determinant can be easily calculated: $$\Delta = \frac{dx}{d\xi}\frac{dy}{d\eta} - \frac{dx}{d\eta}\frac{dy}{d\xi} > 0$$ Where: $$\frac{dx}{d\xi} = \left(\frac{1}{2}-\eta\right)(x_2-x_1) + \left(\frac{1}{2}+\eta\right)(x_4-x_3) \\ \frac{dy}{d\xi} = \left(\frac{1}{2}-\eta\right)(y_2-y_1) + \left(\frac{1}{2}+\eta\right)(y_4-y_3) \\ \frac{dx}{d\eta} = \left(\frac{1}{2}-\xi\right)(x_3-x_1) + \left(\frac{1}{2}+\xi\right)(x_4-x_2) \\ \frac{dy}{d\eta} = \left(\frac{1}{2}-\xi\right)(y_3-y_1) + \left(\frac{1}{2}+\xi\right)(y_4-y_2)$$ Substitution gives: $$\Delta = N_1 \Delta_1 + N_2 \Delta_2 + N_3 \Delta_3 + N_4 \Delta_4$$ Where: $$\Delta_1 = (x_2 - x_1)(y_3 - y_1) - (x_3 - x_1)(y_2 - y_1)\\ \Delta_2 = (x_2 - x_1)(y_4 - y_2) - (x_4 - x_2)(y_2 - y_1)\\ \Delta_3 = (x_4 - x_3)(y_3 - y_1) - (x_3 - x_1)(y_4 - y_3)\\ \Delta_4 = (x_4 - x_3)(y_4 - y_2) - (x_4 - x_2)(y_4 - y_3)$$ Remember the triangle result: $$\Delta = (x_2 - x_1)(y_3 - y_1) - (x_3 - x_1)(y_2 - y_1)$$ Now replace $(1,2,3)$ by $(2,4,1)$ , $(3,1,4)$ , $(4,3,2)$ and respectively attach labels $1$ , $2$ , $3$ and $4$ to these triangles. Then the Jacobians $\Delta_k$ at a quadrilateral are identical to the Jacobians $\Delta_k$ at the triangles labeled $k = 1,2,3,4$ , according to the above. Therefore the Jacobian is expressed by the shape functions into its values $\Delta_k$ at (the triangles beloning to) the nodal points of the quad, as it should be.
Numerical experiments have been performed to see where the Jacobian determinant is $\color{blue}{\mbox{zero}}$, $\color{green}{\mbox{positive}}$ or $\color{red}{\mbox{negative}}$. It turns out that there are also places where the Jacobian is positive and negative, i.e. has more than one value (!) These places have been colored $\color{gray}{\mbox{gray}}$ . Five different cases are distinguished: (1) convex area negative, (2) convex area positive, (3) concave area negative, (4) concave area positive, (5) self-intersecting with positive and negative area:
The local coordinates are such that $\;-1/2<\xi<+1/2,-1/2<\eta<+1/2\;$ and thus extend beyond the quadrilateral for the three degenerated cases (3), (4) and (5) : this is the $\color{gray}{\mbox{gray}}$ area. Careful inspection leads to the following
Conjecture. The Jacobian determinant is $\color{blue}{\mbox{zero}}$ at the curved boundary of the $\color{gray}{\mbox{gray}}$ area.
Two issues:
• Can we prove this conjecture?
• And what is the shape of that curved boundary?
• UPDATE. Look up @ Wikipedia: Jacobian determinant.
A nonlinear map $f: \mathbb{R}^2\to\mathbb{R}^2$ sends a small square to a distorted parallelogram close to the image of the square under the best linear approximation of $f$ near the point. That's what the caption says. $$\vec{r}_1 = \vec{r}(\xi+d\xi,\eta)-\vec{r}(\xi,\eta) = \begin{bmatrix} \large \frac{\partial x}{\partial \xi} \\ \large \frac{\partial y}{\partial \xi} \end{bmatrix} d\xi \\ \vec{r}_2 = \vec{r}(\xi,\eta+d\eta)-\vec{r}(\xi,\eta) = \begin{bmatrix} \large \frac{\partial x}{\partial \eta} \\ \large \frac{\partial y}{\partial \eta} \end{bmatrix} d\eta$$ The area of the distorted parallelogram spanned by the vectors $\vec{r}_1$ and $\vec{r}_2$ is the determinant which is well known from multivariable integration: $$\begin{vmatrix} \large \frac{\partial x}{\partial \xi} & \large \frac{\partial x}{\partial \eta} \\ \large \frac{\partial y}{\partial \xi} & \large \frac{\partial y}{\partial \eta} \end{vmatrix} d\xi\,d\eta = \left(\frac{\partial x}{\partial \xi}\frac{\partial y}{\partial \eta} -\frac{\partial x}{\partial \eta}\frac{\partial y}{\partial \xi}\right) d\xi\,d\eta$$ More pictures, with Jacobian determinant $\color{red}{\mbox{negative}}$ and $\color{green}{\mbox{positive}}$ :
It is seen that the Jacobian determinant always switches sign at the parabolic boundary, which is inevitable because one of the answers below shows that there is only one such boundary, where the determinant is zero. The only way to change sign is to change orientation, because continuity must be ensured, i.e. the quads must remain connected with their neighbor infinitesimal quadrilaterals. This may not be a die hard proof, but it makes me to believe that the Conjecture is indeed true.
• Why was this downvoted? it is a perfectly good question – Rab Sep 15 '17 at 17:27
• @RabMakh: Thank you for this kind remark. – Han de Bruijn Sep 15 '17 at 18:02
A possible clue may be that the interpolation of the Jacobian is not really bi-linear. To see this, take a look at its Finite Difference representation, which is a different way of telling the same as with the Finite Element representation mentioned in the question: $$\Delta = N_1\Delta_1+N_2\Delta_2+N_3\Delta_3+N_4\Delta_4 = A_\Delta + B_\Delta\xi + C_\Delta\eta + D_\Delta\xi\eta$$ Where: $$A_\Delta = \Delta(0,0) = \frac{1}{4}\left(\Delta_1+\Delta_2+\Delta_3+\Delta_4\right)\\ B_\Delta = \frac{\partial \Delta}{\partial \xi}(0,0) = \frac{1}{2}\left(-\Delta_1+\Delta_2-\Delta_3+\Delta_4\right)\\ C_\Delta = \frac{\partial \Delta}{\partial \eta}(0,0) = \frac{1}{2}\left(-\Delta_1-\Delta_2+\Delta_3+\Delta_4\right)\\ D_\Delta = \frac{\partial^2 \Delta}{\partial \xi \partial \eta}(0,0) = \Delta_1-\Delta_2-\Delta_3+\Delta_4$$ The last term $D_\Delta$ is zero. This is seen by considering the fact that the area of the quadrilateral is equal to the sum of the areas of the triangles (1) and (4), but also to the sum of the areas of the triangles (2) and (3):
Therefore $\Delta_1/2 + \Delta_4/2 = \Delta_2/2 + \Delta_3/2$ , which proves the above claim.
We conclude that $\Delta$ is not a bi-linear but only a linear expression in the $\Delta_k$'s. At places where the Jacobian determinant is zero we thus have an equation of the form: $$\Delta = A_\Delta + B_\Delta\xi + C_\Delta\eta = 0 \quad \Longrightarrow \quad C_\Delta\eta = -(A_\Delta + B_\Delta\xi)$$ Now have a look at the Finite Difference representation of the global coordinates: $$\begin{cases} x = A_x + B_x\xi + C_x\eta + D_x\xi\eta\\ y = A_x + B_y\xi + C_y\eta + D_y\xi\eta \end{cases}$$ It follows that, in general, $x$ and $y$ are quadratic functions of the parameter $\xi$ : $$\begin{cases} C_\Delta x &=& C_\Delta A_x + C_\Delta B_x\xi - C_x(A_\Delta + B_\Delta\xi) - D_x\xi(A_\Delta + B_\Delta\xi) \\&=& (C_\Delta A_x - C_x A_\Delta) + (C_\Delta B_x - C_x B_\Delta - D_x A_\Delta)\xi - D_x B_\Delta\xi^2\\ C_\Delta y &=& (C_\Delta A_y - C_y A_\Delta) + (C_\Delta B_y - C_y B_\Delta - D_y A_\Delta)\xi - D_y B_\Delta\xi^2 \end{cases}$$ At Mathematics Stack Exchange we have the following reference to prove that the resulting curve must be a parabola:
Apart from special cases eventually, this answeres the second issue of the question. Free software source confirming this numerically is found at my website . The range of the local coordinates (displayed as $\color{gray}{\mbox{gray}}$ pixels) has been enlarged beyond $[-1/2,+1/2]\times[-1/2,+1/2]$ and (part of) the parabolic curve is drawn in $\color{blue}{\mbox{blue}}$ :
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2021-05-10 08:58:24
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https://socratic.org/questions/how-do-you-write-the-expression-for-the-nth-term-of-the-geometric-sequence-a-4-1
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# How do you write the expression for the nth term of the geometric sequence a_4=-18, a_7=2/3, n=6?
Mar 5, 2017
${a}_{n} = 486 {\left(- \frac{1}{3}\right)}^{n - 1}$
#### Explanation:
$\text{The nth term for a geometric sequence is}$
• a_n=ar^(n-1)" where a is the 1st term"
To obtain the nth term for the given sequence, we require to find a and r.
$\text{Given " a_4=-18" and "a_7=2/3" then}$
$\Rightarrow {a}_{4} = a {r}^{3} = - 18 \to \left(1\right)$
$\Rightarrow {a}_{7} = a {r}^{6} = \frac{2}{3} \to \left(2\right)$
$\Rightarrow \frac{a {r}^{6}}{a {r}^{3}} = \frac{\frac{2}{3}}{- 18}$
$\Rightarrow {r}^{3} = - \frac{1}{27} \Rightarrow \textcolor{red}{r = - \frac{1}{3}}$
$\text{From (1) } a {r}^{3} = - 18$
$\Rightarrow a = \frac{- 18}{- \frac{1}{27}} \Rightarrow \textcolor{red}{a = 486}$
$\Rightarrow \text{nth term expression is } {a}_{n} = 486 {\left(- \frac{1}{3}\right)}^{n - 1}$
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2020-03-31 17:43:39
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http://www.maplesoft.com/support/help/Maple/view.aspx?path=numtheory/cfrac
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numtheory/cfrac(deprecated) - Help
numtheory
cfrac
compute a regular or simple continued fraction expansion
Calling Sequence cfrac(ex_numeric, n, con, den, 'quotients') cfrac(ex_numeric, n, con, den, 'centered', 'quotients') cfrac(ex_numeric, 'periodic', 'quotients') cfrac(ex_rat, x, n, form_rational, 'quotients') cfrac(ex_algebraic, x, n, diag, form_algebraic, 'quotients') cfrac(ex_algebraic, x=a, n, diag, form_algebraic, 'quotients') cfrac(ex_cf)
Parameters
ex_numeric - number (rational, float, algebraic, transcendental, real, or complex) n - (optional) integer (n+1 is the number of partial quotients) con - (optional) unevaluated name (a list of n+1 convergents is assigned) den - (optional) unevaluated name (a list of n+1 denominators is assigned) quotients - (optional) literal name; specifies that cfrac compute the list form (a list of two lists: the first list is the pre-period, the second list is the period) for the continued fraction expansion. centered - (optional) literal name; specifies that cfrac compute a centered form periodic - (optional) literal name; specifies that cfrac compute the simple continued fraction expansion for a quadratic surd ex_numeric ex_rat - rational polynomial x - (optional) unevaluated name; a variable in which the continued fraction is expanded form_rational - (optional) name: simple, regular, or monic ex_algebraic - series or algebraic object diag - (optional) name: superdiagonal or subdiagonal form_algebraic - (optional) name: simple, semisimple, or simregular a - algebraic expression specifying a finite expansion point ex_cf - continued fraction in fraction form or list form
Description
• Important: The numtheory[cfrac] command has been deprecated. Use the superseding command NumberTheory[ContinuedFraction] instead.
• The cfrac command computes a continued fraction expansion of a number, rational polynomial, series, or other algebraic expression.
• There are at least five forms for continued fraction expansions:
1 Regular continued fraction:
$[{b}_{0},[{a}_{1},{b}_{1}],[{a}_{2},{b}_{2}],\mathrm{...},[{a}_{n},{b}_{n}],\mathrm{...}],$
where, usually, ${a}_{i}$ and ${b}_{i}$ for i > 0 are integers or polynomials with integer coefficients. If ${b}_{0}\ne 0$ then it is called superdiagonal, otherwise it is called subdiagonal.
2 Simple continued fraction:
$[{a}_{0},{a}_{1},{a}_{2},\mathrm{...},{a}_{n},\mathrm{...}]=[{a}_{0},[1,{a}_{1}],[1,{a}_{2}],\mathrm{...},[1,{a}_{n}],\mathrm{...}].$
3 Simregular continued fraction:
$[{b}_{0},[{a}_{1},1],[{a}_{2},1],\mathrm{...},[{a}_{n},1],\mathrm{...}].$
4 Semisimple continued fraction:
$[{b}_{0},[{e}_{1},{b}_{1}],[{e}_{2},{b}_{2}],\mathrm{...},[{e}_{n},{b}_{n}],\mathrm{...}],$
where $\left|{e}_{i}\right|=1$, $\mathrm{sign}\left({b}_{i}\right)=1$ for i = 1, 2, ...
5 Monic polynomial continued fraction:
$[{b}_{0},[{e}_{1},{b}_{1}],[{e}_{2},{b}_{2}],\mathrm{...},[{e}_{n},{b}_{n}],\mathrm{...}],$
where each ${b}_{i}$ is a monic polynomial for i = 1, 2, ...
• By default, the cfrac command returns the fraction form of the continued fraction. The form used in the previous illustrations (1)-(5) is referred to as the list form.
The list form for a periodic continued fraction is $\left[\left[{a}_{0},\mathrm{...},{a}_{m}\right],\left[{b}_{1},\mathrm{...},{b}_{n}\right]\right]$, where $0, $1, ${a}_{0}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}∈\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}ℤ$ and ${a}_{1},\dots ,{a}_{n}>0$ describe the preperiod and ${b}_{1},\dots ,{b}_{n}>0$ represent the period, i.e.,
${a}_{0}+\frac{1}{{a}_{1}+\frac{1}{\mathrm{...}+\frac{1}{{a}_{n}+\frac{1}{{b}_{1}+\frac{1}{\mathrm{...}+\frac{1}{{b}_{n}+\frac{1}{{b}_{1}+\frac{1}{\mathrm{...}+\frac{1}{{b}_{n}+\frac{1}{\mathrm{...}}}}}}}}}}}$
• If you specify the quotients option, the continued fraction is returned in list form. For large continued fractions, this form prints more quickly than the fraction form.
• If you specify the n option, at most n + 1 quotients of the continued fraction are computed.
• If cfrac is passed a continued fraction, in fraction or list form, it computes the last convergent of the continued fraction. For a periodic continued fraction, it returns the corresponding quadratic surd.
• The print routine print/CFRAC is used by the prettyprinter to format the fraction form on screen.
• This function is part of the numtheory package, and so can be used in the form cfrac(..) only after performing the command with(numtheory). The function can always be accessed in the long form numtheory[cfrac](..).
• The cfrac command computes a continued fraction expansion for three kinds of input:
6 Numeric
7 Rational polynomials
8 Series or algebraic objects
Numerical Case: Simple and Centered Continued Fraction
• The cfrac(ex_numeric) calling sequence returns a fraction form of the continued fraction for the real or complex number ex_numeric.
If ex_numeric is a complex number, cfrac returns a continued fraction expansion if and only if that expansion is finite. (There are many different definitions of continued fractions for complex numbers.)
• If you specify the con option, cfrac assigns a list containing the convergents to that name.
• If you specify the den option, cfrac assigns a list containing the denominators to that name.
• If you specify the periodic option, cfrac computes a continued fraction for the quadratic surd ex_numeric. None of the other optional arguments (n, con, den) can be specified in this case.
Rational Polynomial Case
• The cfrac(ex_rat) calling sequence returns a fraction form of the continued fraction for the rational polynomial ex_rat.
• You can specify the form of the resulting continued fraction using the form_rational option (simple, regular, or monic). By default, cfrac returns the simple continued fraction form.
• Applying the cfrac command to a finite simple continued fraction (fraction form or list form) returns the original rational polynomial.
Series and Algebraic Object Cases
• The cfrac(ex_algebraic) calling sequence returns a superdiagonal continued fraction approximation for the series or algebraic object ex_algebraic that is equal to the $\left(k,k\right)$ or $\left(k,k-1\right)$ Pade approximant (depending on the parity of the order n if specified).
• Using the cfrac(ex_algebraic, x=a) calling sequence, you can specify an expansion point. The continued fraction will be expressed in terms of powers of $x-a$.
• If you specify the subdiagonal option, cfrac computes the continued fraction equivalent to the $\left(k,k\right)$ or $\left(k-1,k\right)$ Pade approximant.
• You can specify the form of the resulting continued fraction using the form_algebraic option (simple, semisimple, or simregular). By default, cfrac returns the simple continued fraction form.
Examples
> $\mathrm{with}\left(\mathrm{numtheory}\right):$
> $\mathrm{cfrac}\left(\mathrm{π},6\right)$
${3}{+}\frac{{1}}{{7}{+}\frac{{1}}{{15}{+}\frac{{1}}{{1}{+}\frac{{1}}{{292}{+}\frac{{1}}{{1}{+}\frac{{1}}{{1}{+}{\mathrm{...}}}}}}}}$ (1)
> $\mathrm{cfrac}\left(\mathrm{π},100,'\mathrm{quotients}'\right)$
$\left[{3}{,}{7}{,}{15}{,}{1}{,}{292}{,}{1}{,}{1}{,}{1}{,}{2}{,}{1}{,}{3}{,}{1}{,}{14}{,}{2}{,}{1}{,}{1}{,}{2}{,}{2}{,}{2}{,}{2}{,}{1}{,}{84}{,}{2}{,}{1}{,}{1}{,}{15}{,}{3}{,}{13}{,}{1}{,}{4}{,}{2}{,}{6}{,}{6}{,}{99}{,}{1}{,}{2}{,}{2}{,}{6}{,}{3}{,}{5}{,}{1}{,}{1}{,}{6}{,}{8}{,}{1}{,}{7}{,}{1}{,}{2}{,}{3}{,}{7}{,}{1}{,}{2}{,}{1}{,}{1}{,}{12}{,}{1}{,}{1}{,}{1}{,}{3}{,}{1}{,}{1}{,}{8}{,}{1}{,}{1}{,}{2}{,}{1}{,}{6}{,}{1}{,}{1}{,}{5}{,}{2}{,}{2}{,}{3}{,}{1}{,}{2}{,}{4}{,}{4}{,}{16}{,}{1}{,}{161}{,}{45}{,}{1}{,}{22}{,}{1}{,}{2}{,}{2}{,}{1}{,}{4}{,}{1}{,}{2}{,}{24}{,}{1}{,}{2}{,}{1}{,}{3}{,}{1}{,}{2}{,}{1}{,}{1}{,}{10}{,}{2}{,}{\mathrm{...}}\right]$ (2)
> $\mathrm{cfrac}\left({3}^{\frac{1}{2}},'\mathrm{periodic}'\right)$
${1}{+}\frac{{1}}{{1}{+}\frac{{1}}{{2}{+}\frac{{1}}{{1}{+}\frac{{1}}{{2}{+}{\mathrm{...}}}}}}$ (3)
> $\mathrm{cfrac}\left({3}^{\frac{1}{2}},'\mathrm{periodic}','\mathrm{quotients}'\right)$
$\left[\left[{1}\right]{,}\left[{1}{,}{2}\right]\right]$ (4)
> $\mathrm{cfrac}\left(\frac{35470}{99661}+\frac{315I}{99661},'\mathrm{centered}'\right)$
$\frac{{I}}{{3}{}{I}{+}\frac{{1}}{{1}{+}{5}{}{I}{-}\frac{{1}}{{2}{+}{I}{+}\frac{{I}}{{1}{+}{2}{}{I}{-}\frac{{I}}{{3}{+}{2}{}{I}{-}\frac{{I}}{{1}{+}{I}}}}}}}$ (5)
> $f≔\mathrm{cfrac}\left(\frac{{x}^{4}+13{x}^{2}+50x+120}{5{x}^{2}+x+1}\right)$
${f}{:=}\frac{{1}}{{5}}{}{{x}}^{{2}}{-}\frac{{1}}{{25}}{}{x}{+}\frac{{321}}{{125}}{+}\frac{{1}}{\frac{{625}}{{5934}}{}{x}{-}\frac{{2810875}}{{11737452}}{+}\frac{{1}}{\frac{{23216680056}}{{14242671875}}{}{x}{+}\frac{{57431352636}}{{14242671875}}}}$ (6)
> $\mathrm{cfrac}\left(f\right)$
$\frac{{{x}}^{{4}}{+}{13}{}{{x}}^{{2}}{+}{50}{}{x}{+}{120}}{{5}{}{{x}}^{{2}}{+}{x}{+}{1}}$ (7)
> $g≔\mathrm{cfrac}\left({\left(1+x\right)}^{k},x,7,'\mathrm{subdiagonal}','\mathrm{simregular}'\right)$
${g}{:=}\frac{{1}}{{1}{-}\frac{{k}{}{x}}{{1}{+}\frac{{1}}{{2}}{}\frac{\left({k}{+}{1}\right){}{x}}{{1}{-}\frac{{1}}{{6}}{}\frac{\left({k}{-}{1}\right){}{x}}{{1}{+}\frac{{1}}{{6}}{}\frac{\left({k}{+}{2}\right){}{x}}{{1}{-}\frac{{1}}{{10}}{}\frac{\left({k}{-}{2}\right){}{x}}{{1}{+}\frac{{1}}{{10}}{}\frac{\left({k}{+}{3}\right){}{x}}{{1}{+}{\mathrm{...}}}}}}}}}$ (8)
> $\mathrm{op}\left(g\right)$
$\left[{0}{,}\left[{1}{,}{1}\right]{,}\left[{-}{k}{}{x}{,}{1}\right]{,}\left[\frac{{1}}{{2}}{}\left({k}{+}{1}\right){}{x}{,}{1}\right]{,}\left[{-}\frac{{1}}{{6}}{}\left({k}{-}{1}\right){}{x}{,}{1}\right]{,}\left[\frac{{1}}{{6}}{}\left({k}{+}{2}\right){}{x}{,}{1}\right]{,}\left[{-}\frac{{1}}{{10}}{}\left({k}{-}{2}\right){}{x}{,}{1}\right]{,}\left[\frac{{1}}{{10}}{}\left({k}{+}{3}\right){}{x}{,}{1}\right]{,}{\mathrm{...}}\right]$ (9)
> $\mathrm{cfrac}\left({ⅇ}^{x},x=1,5\right)$
${ⅇ}{+}\frac{{x}{-}{1}}{\frac{{1}}{{ⅇ}}{+}\frac{{x}{-}{1}}{{-}{2}{}{ⅇ}{+}\frac{{x}{-}{1}}{{-}\frac{{3}}{{ⅇ}}{+}\frac{{x}{-}{1}}{{2}{}{ⅇ}{+}\frac{{x}{-}{1}}{\frac{{5}}{{ⅇ}}{+}{\mathrm{...}}}}}}}$ (10)
Compatibility
• The optional arguments con and den can be used in conjunction with the option centered as of Maple 16.
• The argument ex_cf can be a periodic continued fraction as of Maple 16.
• The a parameter was introduced in Maple 16.
• The ex_cf parameter was updated in Maple 16.
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2016-08-26 11:57:49
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https://math.stackexchange.com/questions/1847147/prove-that-if-left-x-n-right-converges-then-left-leftx-n-right/1847171
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# Prove that if $\left\{ x_{n}\right\}$ converges then $\left\{ \left(x_{n}\right)^{2}\right\}$ converges.
Good night. I have a problem with this problem. I tried the following:
Proof:
Let $\left\{ x_{n}\right\}$ be a convergent sequence. By definition:
$\mid x_{n}-x\mid<\epsilon$
Then:
$\mid\mid x_{n}\mid-\mid x\mid\mid<\mid x_{n}-x\mid$
$(\mid\mid x_{n}\mid-\mid x\mid\mid)^{2}<(\mid x_{n}-x\mid)^{2}$
$\mid(\mid x_{n}\mid-\mid x\mid)^{2}\mid<(\mid x_{n}-x\mid)^{2}$
$\mid\mid x_{n}\mid^{2}-\mid\left(x_{n}\right)\left(x\right)\mid+\mid x\mid^{2}\mid<(\mid x_{n}-x\mid)^{2}$
Suppose that $x_n\to x$, for some $x\in\mathbb{R}$. Also suppose you have $\epsilon>0$.
Now, because $\{x_n\}$ is a convergent sequence we have that ${x_n}$ is bounded and so $\exists M\in\Bbb {R^+}:\forall n\in\Bbb N \ \ |x_n|\le M$ (and so $x\le M$) and because $x_n\to x$ $\exists n_0\in\Bbb N:\forall n\geq n_0 \ \ |x_n-x|<\frac{\epsilon}{2M}.$
Then, for all $n\geq n_0$ we have that $|(x_n)^2-x^2|=|x_n-x||x_n+x|\le (|x_n|+|x|)|x_n-x|\le 2M|x_n-x|\le 2M\cfrac{\epsilon}{2M}=\epsilon$
and so $(x_n)^2\to x^2$.
• +1 This was the exact proof I had already lined up myself. You beat me by seconds! – NoseKnowsAll Jul 2 '16 at 22:08
• Thanks man, but i don't understand the $2M\mid x_{n}-x\mid\leq2M\frac{\epsilon}{2M}=\epsilon$ why e/2m? – Bvss12 Jul 3 '16 at 1:56
• Yeah. my wrong, check again the edit. – richarddedekind Jul 3 '16 at 13:04
Hint: if $a=\lim_{n\to \infty}a_n$ exists and $b=\lim_{n\to \infty}b_n$ exists, then $\lim_{n\to \infty}a_nb_n$ exists and equals $ab$.
• I think a delta-epsilon hint would be more useful. – ReverseFlow Jul 2 '16 at 21:31
• @ReverseFlow I think using the algebra of limits when applicable saves a lot of work and also sheds more light into the underlying structure. While a delta-epsilon approach is mostly always tailored to a particular example, this kind of technique can be used in much broader settings. – Fimpellizieri Jul 2 '16 at 21:35
• I agree. Based on his work it seems that the delta-epsilon approach is what is being asked. If he is allowed to use lemmas, like the ones you mention, then there really isn't much to do. – ReverseFlow Jul 2 '16 at 21:38
• Yeah, but your hint is as fundamental as the problem above – richarddedekind Jul 2 '16 at 21:39
We have $$\forall { \varepsilon }_{ 1 }>0\quad ,\exists { n }_{ { \varepsilon }_{ 1 } },n\ge { n }_{ { \varepsilon }_{ 1 } }\quad \mid x_{ n }-x\mid <{ \varepsilon }_{ 1 }$$ and we have to show $$\lim _{ n\rightarrow \infty }{ { x }_{ n }^{ 2 } } =a$$ where $a$ is equal to $x=\sqrt { a } ,$ $$\quad \mid { x }_{ n }^{ 2 }-a\mid =\left| \left( { x }_{ n }+\sqrt { a } \right) \left( { x }_{ n }-\sqrt { a } \right) \right| =\left| { x }_{ n }+\sqrt { a } \right| \left| { x }_{ n }-\sqrt { a } \right| <{ \varepsilon }_{ 1 }\left| { x }_{ n }+\sqrt { a } \right| =\\ ={ \varepsilon }_{ 1 }\left| { { x }_{ n }-\sqrt { a } +2\sqrt { a } } \right| <{ \varepsilon }_{ 1 }\left| { x }_{ n }-\sqrt { a } \right| +2={ \varepsilon }_{ 1 }^{ 2 }+2\left| { \varepsilon }_{ 1 }\sqrt { a } \right|$$ and ${ \varepsilon }_{ 2 }={ \varepsilon }_{ 1 }^{ 2 }+2\left| { \varepsilon }_{ 1 }\sqrt { a } \right|$ so we have $\forall { \varepsilon }_{ 2 }>0\quad ,\exists { n }_{ { \varepsilon }_{ 2 } },n\ge { n }_{ { \varepsilon }_{ 2 } }$
$$|{ x }_{ n }^{ 2 }-a\mid <{ \varepsilon }_{ 2 }$$
Suppose that the sequence $\{x_n\}$ converges to $a$. We claim that the squared sequence converges to $a^2$. Note that $$|x_n^2-a^2|=|x_n-a||x_n+a|.$$ We bound $|x_n+a|$ by requiring that $|x_n-a|<1$ so that $|x_n+a|=|x_n-a+2a|<1+2|a|$ . Given $\epsilon>0$ choose $N$ s.t. $n\geq N$ implies that $|x_n-a|<\min(1, \epsilon/(1+2|a|))$ and the result follows.
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2019-05-19 20:38:09
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http://wikien4.appspot.com/wiki/Transpose
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# Transpose
The transpose AT of a matrix A can be obtained by refwecting de ewements awong its main diagonaw. Repeating de process on de transposed matrix returns de ewements to deir originaw position, uh-hah-hah-hah.
In winear awgebra, de transpose of a matrix is an operator which fwips a matrix over its diagonaw; dat is, it switches de row and cowumn indices of de matrix A by producing anoder matrix, often denoted by AT (among oder notations).[1][2]
The transpose of a matrix was introduced in 1858 by de British madematician Ardur Caywey.[3]
## Transpose of a matrix
### Definition
The transpose of a matrix A, denoted by AT,[1][4] A, A, A′,[5] Atr, tA or At, may be constructed by any one of de fowwowing medods:
1. Refwect A over its main diagonaw (which runs from top-weft to bottom-right) to obtain AT;
2. Write de rows of A as de cowumns of AT;
3. Write de cowumns of A as de rows of AT.
Formawwy, de i-f row, j-f cowumn ewement of AT is de j-f row, i-f cowumn ewement of A:
${\dispwaystywe \weft[\madbf {A} ^{\operatorname {T} }\right]_{ij}=\weft[\madbf {A} \right]_{ji}.}$
If A is an m × n matrix, den AT is an n × m matrix.
In de case of sqware matrices, AT may awso denote de Tf power of de matrix A. For avoiding a possibwe confusion, many audors use weft upperscripts, dat is, dey denote de transpose as TA. An advantage of dis notation is dat no parendeses are needed when exponents are invowved: as (TA)n = T(An), notation TAn is not ambiguous.
In dis articwe dis confusion is avoided by never using de symbow T as a variabwe name.
#### Matrix definitions invowving transposition
A sqware matrix whose transpose is eqwaw to itsewf is cawwed a symmetric matrix; dat is, A is symmetric if
${\dispwaystywe \madbf {A} ^{\operatorname {T} }=\madbf {A} .}$
A sqware matrix whose transpose is eqwaw to its negative is cawwed a skew-symmetric matrix; dat is, A is skew-symmetric if
${\dispwaystywe \madbf {A} ^{\operatorname {T} }=-\madbf {A} .}$
A sqware compwex matrix whose transpose is eqwaw to de matrix wif every entry repwaced by its compwex conjugate (denoted here wif an overwine) is cawwed a Hermitian matrix (eqwivawent to de matrix being eqwaw to its conjugate transpose); dat is, A is Hermitian if
${\dispwaystywe \madbf {A} ^{\operatorname {T} }={\overwine {\madbf {A} }}.}$
A sqware compwex matrix whose transpose is eqwaw to de negation of its compwex conjugate is cawwed a skew-Hermitian matrix; dat is, A is skew-Hermitian if
${\dispwaystywe \madbf {A} ^{\operatorname {T} }=-{\overwine {\madbf {A} }}.}$
A sqware matrix whose transpose is eqwaw to its inverse is cawwed an ordogonaw matrix; dat is, A is ordogonaw if
${\dispwaystywe \madbf {A} ^{\operatorname {T} }=\madbf {A} ^{-1}.}$
A sqware compwex matrix whose transpose is eqwaw to its conjugate inverse is cawwed a unitary matrix; dat is, A is unitary if
${\dispwaystywe \madbf {A} ^{\operatorname {T} }={\overwine {\madbf {A} ^{-1}}}.}$
### Exampwes
• ${\dispwaystywe {\begin{bmatrix}1&2\end{bmatrix}}^{\operatorname {T} }=\,{\begin{bmatrix}1\\2\end{bmatrix}}}$
• ${\dispwaystywe {\begin{bmatrix}1&2\\3&4\end{bmatrix}}^{\operatorname {T} }={\begin{bmatrix}1&3\\2&4\end{bmatrix}}}$
• ${\dispwaystywe {\begin{bmatrix}1&2\\3&4\\5&6\end{bmatrix}}^{\operatorname {T} }={\begin{bmatrix}1&3&5\\2&4&6\end{bmatrix}}}$
### Properties
Let A and B be matrices and c be a scawar.
1. ${\dispwaystywe \weft(\madbf {A} ^{\operatorname {T} }\right)^{\operatorname {T} }=\madbf {A} .}$
The operation of taking de transpose is an invowution (sewf-inverse).
2. ${\dispwaystywe \weft(\madbf {A} +\madbf {B} \right)^{\operatorname {T} }=\madbf {A} ^{\operatorname {T} }+\madbf {B} ^{\operatorname {T} }.}$
3. ${\dispwaystywe \weft(\madbf {AB} \right)^{\operatorname {T} }=\madbf {B} ^{\operatorname {T} }\madbf {A} ^{\operatorname {T} }.}$
Note dat de order of de factors reverses. From dis one can deduce dat a sqware matrix A is invertibwe if and onwy if AT is invertibwe, and in dis case we have (A−1)T = (AT)−1. By induction, dis resuwt extends to de generaw case of muwtipwe matrices, where we find dat (A1A2...Ak−1Ak)T = AkTAk−1TA2TA1T.
4. ${\dispwaystywe \weft(c\madbf {A} \right)^{\operatorname {T} }=c\madbf {A} ^{\operatorname {T} }.}$
The transpose of a scawar is de same scawar. Togeder wif (2), dis states dat de transpose is a winear map from de space of m × n matrices to de space of aww n × m matrices.
5. ${\dispwaystywe \det \weft(\madbf {A} ^{\operatorname {T} }\right)=\det(\madbf {A} ).}$
The determinant of a sqware matrix is de same as de determinant of its transpose.
6. The dot product of two cowumn vectors a and b can be computed as de singwe entry of de matrix product:
${\dispwaystywe \weft[\madbf {a} \cdot \madbf {b} \right]=\madbf {a} ^{\operatorname {T} }\madbf {b} ,}$
which is written as aibi in Einstein summation convention.
7. If A has onwy reaw entries, den ATA is a positive-semidefinite matrix.
8. ${\dispwaystywe \weft(\madbf {A} ^{\operatorname {T} }\right)^{-1}=\weft(\madbf {A} ^{-1}\right)^{\operatorname {T} }.}$
The transpose of an invertibwe matrix is awso invertibwe, and its inverse is de transpose of de inverse of de originaw matrix. The notation A−T is sometimes used to represent eider of dese eqwivawent expressions.
9. If A is a sqware matrix, den its eigenvawues are eqwaw to de eigenvawues of its transpose, since dey share de same characteristic powynomiaw.
### Products
If A is an m × n matrix and AT is its transpose, den de resuwt of matrix muwtipwication wif dese two matrices gives two sqware matrices: A AT is m × m and AT A is n × n. Furdermore, dese products are symmetric matrices. Indeed, de matrix product A AT has entries dat are de inner product of a row of A wif a cowumn of AT. But de cowumns of AT are de rows of A, so de entry corresponds to de inner product of two rows of A. If pi j is de entry of de product, it is obtained from rows i and j in A. The entry pj i is awso obtained from dese rows, dus pi j = pj i, and de product matrix (pi j) is symmetric. Simiwarwy, de product AT A is a symmetric matrix.
A qwick proof of de symmetry of A AT resuwts from de fact dat it is its own transpose:
${\dispwaystywe \weft(\madbf {A} \madbf {A} ^{\operatorname {T} }\right)^{\operatorname {T} }=\weft(\madbf {A} ^{\operatorname {T} }\right)^{\operatorname {T} }\madbf {A} ^{\operatorname {T} }=\madbf {A} \madbf {A} ^{\operatorname {T} }.}$[6]
### Impwementation of matrix transposition on computers
Iwwustration of row- and cowumn-major order
On a computer, one can often avoid expwicitwy transposing a matrix in memory by simpwy accessing de same data in a different order. For exampwe, software wibraries for winear awgebra, such as BLAS, typicawwy provide options to specify dat certain matrices are to be interpreted in transposed order to avoid de necessity of data movement.
However, dere remain a number of circumstances in which it is necessary or desirabwe to physicawwy reorder a matrix in memory to its transposed ordering. For exampwe, wif a matrix stored in row-major order, de rows of de matrix are contiguous in memory and de cowumns are discontiguous. If repeated operations need to be performed on de cowumns, for exampwe in a fast Fourier transform awgoridm, transposing de matrix in memory (to make de cowumns contiguous) may improve performance by increasing memory wocawity.
Ideawwy, one might hope to transpose a matrix wif minimaw additionaw storage. This weads to de probwem of transposing an n × m matrix in-pwace, wif O(1) additionaw storage or at most storage much wess dan mn. For n ≠ m, dis invowves a compwicated permutation of de data ewements dat is non-triviaw to impwement in-pwace. Therefore, efficient in-pwace matrix transposition has been de subject of numerous research pubwications in computer science, starting in de wate 1950s, and severaw awgoridms have been devewoped.
## Transposes of winear maps and biwinear forms
Recaww dat matrices can be pwaced into a one-to-one correspondence wif winear operators. The transpose of a winear operator can be defined widout any need to consider a matrix representation of it. This weads to a much more generaw definition of de transpose dat can be appwied to winear operators dat cannot be represented by matrices (e.g. invowving many infinite dimensionaw vector spaces).
### Transpose of a winear map
Let X# denote de awgebraic duaw space of an R-moduwe X. Let X and Y be R-moduwes. If u : XY is a winear map, den its awgebraic adjoint or duaw,[7] is de map #u : Y#X# defined by ffu. The resuwting functionaw u#(f) is cawwed de puwwback of f by u. The fowwowing rewation characterizes de awgebraic adjoint of u[8]
u#(f), x⟩ = ⟨f, u(x)⟩ for aww fY' and xX
where ⟨•, •⟩ is de naturaw pairing (i.e. defined by z, h⟩ := h(z)). This definition awso appwies unchanged to weft moduwes and to vector spaces.[9]
The definition of de transpose may be seen to be independent of any biwinear form on de moduwes, unwike de adjoint (bewow).
The continuous duaw space of a topowogicaw vector space (TVS) X is denoted by X'. If X and Y are TVSs den a winear map u : XY is weakwy continuous if and onwy if u#(Y') ⊆ X', in which case we wet tu : Y'X' denote de restriction of u# to Y'. The map tu is cawwed de transpose[10] of u.
If de matrix A describes a winear map wif respect to bases of V and W, den de matrix AT describes de transpose of dat winear map wif respect to de duaw bases.
### Transpose of a biwinear form
Every winear map to de duaw space u : XX# defines a biwinear form B : X × XF, wif de rewation B(x, y) = u(x)(y). By defining de transpose of dis biwinear form as de biwinear form tB defined by de transpose tu : X##X# i.e. tB(y, x) = tu(Ψ(y))(x), we find dat B(x, y) = tB(y, x). Here, Ψ is de naturaw homomorphism XX## into de doubwe duaw.
If de vector spaces X and Y have respectivewy nondegenerate biwinear forms BX and BY, a concept known as de adjoint, which is cwosewy rewated to de transpose, may be defined:
If u : XY is a winear map between vector spaces X and Y, we define g as de adjoint of u if g : YX satisfies
${\dispwaystywe B_{V}{\big (}x,g(y){\big )}=B_{W}{\big (}u(x),y{\big )}}$ for aww xX and yY.
These biwinear forms define an isomorphism between X and X#, and between Y and Y#, resuwting in an isomorphism between de transpose and adjoint of u. The matrix of de adjoint of a map is de transposed matrix onwy if de bases are ordonormaw wif respect to deir biwinear forms. In dis context, many audors use de term transpose to refer to de adjoint as defined here.
The adjoint awwows us to consider wheder g : YX is eqwaw to u −1 : YX. In particuwar, dis awwows de ordogonaw group over a vector space X wif a qwadratic form to be defined widout reference to matrices (nor de components dereof) as de set of aww winear maps XX for which de adjoint eqwaws de inverse.
Over a compwex vector space, one often works wif sesqwiwinear forms (conjugate-winear in one argument) instead of biwinear forms. The Hermitian adjoint of a map between such spaces is defined simiwarwy, and de matrix of de Hermitian adjoint is given by de conjugate transpose matrix if de bases are ordonormaw.
## References
1. ^ a b "Comprehensive List of Awgebra Symbows". Maf Vauwt. 2020-03-25. Retrieved 2020-09-08.
2. ^ Nykamp, Duane. "The transpose of a matrix". Maf Insight. Retrieved September 8, 2020.
3. ^ Ardur Caywey (1858) "A memoir on de deory of matrices", Phiwosophicaw Transactions of de Royaw Society of London, 148 : 17–37. The transpose (or "transposition") is defined on page 31.
4. ^ T.A. Whitewaw (1 Apriw 1991). Introduction to Linear Awgebra, 2nd edition. CRC Press. ISBN 978-0-7514-0159-2.
5. ^ Weisstein, Eric W. "Transpose". madworwd.wowfram.com. Retrieved 2020-09-08.
6. ^ Giwbert Strang (2006) Linear Awgebra and its Appwications 4f edition, page 51, Thomson Brooks/Cowe ISBN 0-03-010567-6
7. ^ Schaefer & Wowff 1999, p. 128.
8. ^ Hawmos 1974, §44
9. ^ Bourbaki 1989, II §2.5
10. ^ Trèves 2006, p. 240.
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2021-01-23 11:31:58
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http://mathoverflow.net/revisions/43381/list
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MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).
2 removed trivial sign issue
More precisely, what real numbers $r$ have the following property: for any $\epsilon > 0$ there exist infinitely many pairs $(p, q)$ of positive integers such that
$$\left| \frac{p}{q} - r \right| < \frac{\epsilon}{q^2}.$$
I think that this is impossible if $r$ is a quadratic irrational. On the other hand, it's certainly possible for any number with irrationality measure strictly greater than $2$.
What I really want to know is if the real numbers which don't have the property above have measure zero. If that's true, it would answer the last part of this math.SE question.
1
# What numbers can be approximated "pretty well" by rationals?
More precisely, what real numbers $r$ have the following property: for any $\epsilon > 0$ there exist infinitely many pairs $(p, q)$ of positive integers such that
$$\left| \frac{p}{q} - r \right| < \frac{\epsilon}{q^2}.$$
I think that this is impossible if $r$ is a quadratic irrational. On the other hand, it's certainly possible for any number with irrationality measure strictly greater than $2$.
What I really want to know is if the real numbers which don't have the property above have measure zero. If that's true, it would answer the last part of this math.SE question.
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2013-06-18 23:46:57
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https://socratic.org/questions/how-do-functional-groups-affect-organic-molecules
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How do functional groups affect organic molecules?
Sep 24, 2015
Functional groups determine the identity of organic molecules
Explanation:
Functional groups are very important in organic chemistry since they determine the identity of these molecules and therefore, depict the chemical and physical properties of them.
For example, hydroxy groups $O H$ is specific for alcohols. the general structure for alcohols is $R - O H$.
In a similar way, $C O O H$ is the functional group for carboxylic acids of a general structure $R - C O O H$.
For esters, the functional group is $C O O R '$, where esters have the general structure $R - C O O R '$.
And so on.
with different functional groups, alcohols, carboxylic acids, and esters have different chemical and physical properties and therefore, different identity.
Let us take a specific example; Propane $C {H}_{3} C {H}_{2} C {H}_{3}$. Propane is a gaseous molecule at room temperature. Adding a hydroxy group to its structure we get $C {H}_{3} C {H}_{2} C {H}_{2} - O H$ which is the propanol. Propanol is liquid at room temperature ( change on physical property ). Propanol can react thanks to the presence of the oxygen atom in its backbone, however, propane has very limited reactivity ( change on chemical property ).
There are many other functional groups such as:
Sulfides: $R - S - R '$
Thiols: $R - S H$
Ethers: $R - O - R '$
Anhydrides: $R - C O - O - C O - R '$
Ketones: $R - C O - R '$
Aldehydes: $R - C O - H$
Amides: $R - C O - N {\left(R '\right)}_{2}$ or $R - C O - N {H}_{2}$ or $R - C O - N H R '$
Amines: $R - N {H}_{2}$ or $R - N H R '$ or $R - N {\left(R '\right)}_{2}$
Nitriles: $R - C N$
Alkyl halides: $R - X$ where $X$ is an element from group 17.
Alkenes: $R - C H = C H - R '$
Alkynes: $R - C \equiv C - R '$
etc.
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2020-09-28 16:42:59
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http://www.ilovestats.org/the-binomial-distribution/
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## The binomial distribution
I think the binomial distribution is the most fun of all distributions. In this section you will need your skills in probability theory. Here I elaborate on the example about Heads and Tails. The binomial distribution can for example be used to check if a coin really is fair; is the probability of a Head really 0.5?
In the binomial distribution you only need to be concerned about two outcomes; Head or Tail, yes or no, present or absent, 1 or 0. From every single toss of a coin, you can have the outcome of either a Head or a Tail. Every time you ask someone if he or she likes broccoli you can get either yes or no. This variable is therefore discrete; it is on the nominal scale. Head or Tail does not have an internal order.
The binomial distribution id defined by the following equation:
where $P$ is the probability of getting a combination of a specific outcome $x$ times in $k$ trials, $p$ is the probability of the outcome and $q = 1-p$ =is the probability of the other outcome.
This equation looks quite intimidating at a first glance, but it is quite straightforward actually. What the equation does is to calculate the probability of getting a combination with a specific number of outcomes on a specific number of trials.
Example
If you toss a coin 3 times, what is the probability of getting 2 Heads when the probability of getting a Head is 0.5?
Answer: The probability is 0.375 to get a combination with 2 Heads in 3 tosses when p =0.5.
How to do it in R
```dbinom(2,3,prob=0.5)
```
Below I present the binomial distribution for 10 trials and p = 0.5:
When p = 0.5 the distribution is symmetric, but see what happens when p=0.2:
Now, the distribution is asymmetrical. Thus, the distribution only conforms to a perfect normal bell-shaped distribution when p =0.5.
### Important to remember:
In the binomial distribution you are dealing with two outcomes that you can observe from a trial.
You can use the binomial distribution to answer the following question:
“I know the probability getting a Head with my coin is 0.5, what is the probability of getting 2 Heads in 3 tosses?”
### More in depth
How does the equation for the binomial distribution work?
Ok, now let’s go on to look at some probabilities.
The probability of getting a Head in every toss is p= 0.5 and the probability of getting a Tail is 1-p=q= 0.5.
When tossing the coin, say, three times you can get one of the eight following observations:
HHH = 3 Heads
TTT = 3 Tails = 0 Heads
HHT = 1 Tail = 2 Heads
HTH = 1 Tail = 2 Heads
THH = 1 Tail = 2 Heads
TTH = 2 Tails = 1 Head
THT = 2 Tails = 1 Head
HTT = 2 Tails = 1 Head
The probability of each of these combinations is:
Do you see a pattern? If you are not concerned with in which order the Head and Tail comes in the three tosses there are really only four unique combinations:
So the probability of getting 3, 0, 2 or 1 Head in three tosses is equal to the sum of the probability of all combinations containing 3, 0, 2 or 1 Head:
To simplify this table:
But how are we calculating the number of combinations with $x$ number of Heads. We have just dealt with 3 tosses now, but what if we are tossing the coin 10 times. How many combinations do we get for 3 Heads? The answer is 120. And for 100 tosses? The answer is 161 700. From this it is clear that we don’t have time to make tables even when dealing with a number of tosses as low as 10.
So to calculate the probability of a specific number of Heads (X) in k tosses we are multiplying the number of combinations (C) containing that number of Heads with the probability of each of these combinations (C x pC). Do you follow? We are using one of the principles from the probability theory here; we want the probability for a combination containing 2 Heads. There are three combinations with 2 heads, so we add them: 0.125 + 0.125 + 0.125 = 0.125 x 3 = 0.375. The probability for all possible combinations adds up to 1: 0.125 + 0.375 + 0.375 + 0.125 = 1.
Now we are on the way to an equation for calculating the probability of X number of Heads in k tosses. Simplified this equation is P (X, k) = C x pc
To calculate the number of combinations ($C$) with $x$ number of a specific outcome (e.g. Heads) on $k$ number of trials (tosses) you use the following equation:
Example
Calculate the number of combinations you can get containing 6 Heads on 20 tosses.
$k$= 20, $x$= 6
Use the equation:
The answer is: You can get 38 760 combinations with 6 Heads on 20 tosses
Ok, so now we have an equation to calculate the number of combinations (C). But how do we calculate the probability of each combination? This is the most straightforward part. You take the probability of the outcome (e.g. p = 0.5) and raise it to the power of the number of times you will receive the outcome (e.g. 3 times): $p^x$But for the rest of the tosses( $k-x$), you will receive the other outcome ($q$).
On $k$ tosses you use the following equation:
where $P_c$ is the probability of one combination with $x$ nr of heads (or 1:s) in $k$ tosses, $p$ is the probability of getting a head (or 1) and $q$ is the probability of getting a tail (or 0).
Example
Calculate the probability for each combination containing 6 Heads in 20 tosses where the probability of getting a Head in each toss is p = 0.5 and a Tail is q = 1- p = 0.5
$k$= 20, $x$ = 6
Use the equation:
Answer: The probability for each combination containing 6 Heads in 20 tosses is 9.53 x 10-7
Now, we are getting somewhere. Then we combine these to equations (C and Pc). That is the equation for the binomial distribution:
where $P$ is the probability of getting a combination of a specific outcome $x$ times in $k$ trials, $p$ is the probability of the outcome and $q$= 1 – $p$ is the probability of the other outcome.
Example
Calculate the probability of getting a combination with 6 Heads in 20 tosses where the probability of a Head is p=0.5.
$k$= 20, $x$= 6
Use the equation for the binomial distribution:
Answer: The probability of getting a combination with 6 Heads on 20 tosses is 0.037.
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2018-05-22 15:33:19
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https://gamedev.stackexchange.com/questions/133565/directx-12-and-feature-levels
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# DirectX 12 and Feature levels
As far as I know the DirectX12 SDK that comes with the Windows 10 SDK can only be used on a Windows 10 machine. Although I'm not entirely sure about the runtime and the use of future levels (which came with DX11). Considering the radical API changes in DX12, is it possible to initiate a level 11_0 device that will operate on the DX11 runtime on Windows 7 for instance? Or am I getting the feature levels idea wrong?
You are confusing the "Runtime version" with the Direct3D Hardware Feature level. They are not the same thing.
The DirectX 12 API is only supported by the Windows 10 operating system. There is no update for older versions of Windows that would install the DirectX 12 API.
If you need to support Windows 7, either use Direct3D 11 only or provide an alternative codepath that uses Direct3D 11.
The DirectX 12 API supports Feature Level 11.0, 11.1, 12.0, and 12.1 hardware. This also requires WDDM 2.0 drivers. Technically it would be possible for a video card manufacturer to make a driver for DirectX 12 that supported older Direct3D hardware feature levels, but there isn't one or likely to be one.
If you need to support a broad range of video hardware, you should stick with Direct3D 11 which supports 9.1, 9.2, 9.3, 10.0, 10.1, 11.0, or later. Windows 8 or later is required to support Direct3D hardware feature level 11.1, and Windows 10 is required to support Direct3D hardware feature level 12.0 or 12.1.
Any machine that can run DirectX 12 can run your program.
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2019-11-14 22:40:03
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https://forum.eveuniversity.org/viewtopic.php?f=186&t=110121&p=891666
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## [WAR] Strike on Luminaire
Member
### [WAR] Strike on Luminaire
Participants
SPOILER WARNING!
Strontium Bob BobbinsMagnar ErikksonAres DodekatheonMols HallecklobevaserKris Hawk (OOC Scout)Jacob Velora
Fleet Preperation
We recieved intelligence of an enemy Gnosis, Vexor, Exequror and Garmur at the Federation Grand Prix beacon in Luminaire.
I prepared a fleet of 2 x BNI, 1 caracal, 1 exequror and 1 Vexor knowing that they could upship (A Hurricane Fleet Issue and a Vedmak had previously been spotted). The BNIs were to engage the Gnosis and Vexor in close combat. The Vexor and Caracal were for ranged engagements.
Fleet Actions
The OOC Scout confirmed their presence in system but the Gnosis had unfortunately left the field.
Strontium Bob warped to the site and tackled the Vexor. Once tackle was confirmed we jumped in and assisted Bob. I ordered drones and the Caracal to engage the enemy Exequror which forced him off field. The Vexor was then easily destroyed. The Garmur had burned out of range upon seeing our fleet and was too fast to catch so he escaped (but he was never going to be primary using BNI tackle).
Losses:
None
Kills:
1 x Vexor https://zkillboard.com/kill/70754974/ 28M ISK
Thanks to everyone who attended.
Member
### Re: [WAR] Strike on Luminaire
Nice job, saw them as well yesterday, wondering if I should report them, but didn't know where to put the information in anyways, so I didn't.
I even got tackled by their Vexor and their Daredevil, but for some reason they didn't had a scram. So, I heated my MWD and got out of their webs, before they could burn my shields down.
Glad to see you got a kill out of them. o7
Member
### Re: [WAR] Strike on Luminaire
Nice job Jacob and all for engaging wartargets at their hunting grounds! As HSC we are trying to become more and more proactive against wartargets, but this war with BLACKHOGS. is especially proved that we can also stage small size fleets to hunt them not only in our home region but at different regions.
Alexander you can post wartarget activity in intel.
"In Blasters I Trust"
Member
### Re: [WAR] Strike on Luminaire
Nice, thanks Ersin. Completely, forgot about the in-game channel! Going to re-find my way into intel.
Member
### Re: [WAR] Strike on Luminaire
Alexander Oromov wrote:Nice, thanks Ersin. Completely, forgot about the in-game channel! Going to re-find my way into intel.
Good idea, when you move on from the Uni your Alliance's (or coalition's) intel channel will be your best friend, in most cases
I'm not super up to date on this particular war but this sounds like fun and you killed things and didn't die - well done!
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2019-07-20 01:30:46
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http://rwjt.auus.pw/how-to-find-supply-and-demand-equations.html
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# How To Find Supply And Demand Equations
Suppose the supply and demand for a certain textbook are given by supply p+1/3q^2, demand =-1/3q^2+40, where p is price and q is quantity. Inverse demand equation. Substituting P = 5 back into either the supply or the demand equation and solving for Q, Q = 50. quantity is Q*. 80 per bushel, the annual supply for soybeans in the Midwest is 1. The equations will take the form of y=mx+b, where m = slope of the line and b = y-intercept of the line. 3: Business Applications of Quadratic Functions Supply, Demand and Market Equilibrium SAME as Sec. Find This Content on the Updated Version of this Blog: whystudyeconomics. demand and supply on the world markets as −0. If the supply is more elastic than demand, then the producer will bear a greater burden of the tax. Answering the following questions using equations: a. Market Demand. A labor supply curve shows the number of workers who are willing and able to work in an occupation at different wages. It postulates that, holding all else equal, in a competitive market, the unit price for a particular good, or other traded item such as labor or liquid financial assets, will vary until it settles at a point where the quantity demanded (at the current price) will equal the quantity supplied (at the. Introduction to Demand and Supply curves. 20 per bushel,the daily supply decreases to 350 bushels,and the daily demand increases to 980 bushels. by Kenneth Matziorinis. Before you enter into an agreement, it is important to comprehend the conditions and conditions of the arrangement. The demand equation can be written as a regular linear equation and an inverse equation. purchase order quantities). i need details on HOW to solve it. The demand curve shows the amount of goods consumers are willing to buy at each market price. To get the base, find equilibrium quantity (Q*). Algebra -> Quadratic Equations and Parabolas -> SOLUTION: The demand and supply equations for a certain item are given by D = –5p + 40 S = –p2 + 30p – 8 Find the equilibrium price. For the pair of supply-and-demand equations, where x represents the quantity demanded in units of 1,000 and p is the unit price in dollars, find the equilibrium quantity and the equilibrium price. Because demand can be represented graphically as a straight line with price on the y-axis and quantity on the x-axis, a demand equation can be as basic as a linear equation. The market supply curve is the horizontal sum of all individual supply curves. In a perfect market the supply curve is Qs=9p and the demand curve is Qd=100-p. Elasticity of Demand and Supply # 10. Assume that supply and demand equations are linear. Systems of linear equations can be used to solve resource allocation problems in business and economics (see Problems 73 and 76 in Section 4. The demand equation for a certain type of printer is given by: D=-200p + 35,000 The supply equation is predicted to be: s= -p^2 + 400p - 20,000 Find the equilibrium price Do I solve for p first in the demand equation, and then input the answer into the supply equation? I did that and then would get: s = -175^2 + 400(175) -20,000 Am I doing this. Determine QD and QS. For problems 10-12, given the supply and demand data: Find equations of the supply and demand curves, assuming they are both linear. Figure 2: Trade Market for Beef Between California and Nebraska Q P XS n MD c 15 3 11 160 (b) Now consider Nebraska, with the following demand and supply schedules for beef: D n = 100−5P S n = 40+15P. This implies that. Identifying High Quality Supply and Demand Levels By Joe Wright | May 20, 2014 Supply and demand levels can be found on any and all price charts, on every time frame for any instrument, but not all SD levels are created equal. if the price is $1200, then the quantity demanded is 40. For each decrease in unit price of$20 below $485, the quantity demanded increases by 250 units. These equations simply represent the relationship between price and quantity in 'maths language'. This approach is unwise because the latter profoundly impacts the former. Assume that the supply and demand equations are linear. 1Qd and P = 5 +. By using these determinants, businesses can estimate how a change in the price affects demand. B)the supply curve of a normal good shifts rightward. To determine the point price elasticity of demand given P 0 is$1. We will cover the basic definition of an exponential function, the natural exponential function, i. Qs = -4+2P. Solve the equation by means of the quadratic formula where a = 231, b = -20, and c = -4. Supply and demand graph template to quickly visualize demand and supply curves. equilibrium quantity 12,000 units; equilibrium price $9. 3 on production schedules for boats and leases for airplanes). Besides altering the equilibrium price, which takes demand into account, sales tax. Qd = 66-3P. What we're going to think about in this video is elasticity of demand-- tis-sit-tity, elasticity of demand. A demand equation or a exact function expresses demand q (the number of items demanded) as a function of the unit price p (the price per item). We find that the aggregate supply curve is flat enough for the structural demand shock to have important short-run effects on output. p = 80 - 3 x 2 and p = x 2 + 4 x + 32 a) Find the equilibrium quantity (in thousand units). Both intersect at point E and determine OP equilibrium price and OQ quantity demanded and supplied. A new video on Linear Supply equations. Shifting of Demand and Supply. In microeconomics, supply and demand is an economic model of price determination in a market. Assume that steel has linear demand and supply curves throughout. balancing equations practice An agreement doesn’t necessarily have to get placed in writing and can be created through verbal negotiations. by Kenneth Matziorinis. The effect of the subsidy has been to increase demand (shift the. P(s 2) = P(d 2) Part B. identify the supply curve, d. Go through all the prices P = 1 through P = 10 in this fashion, and you'll have plotted out the demand curve. These equations correspond to the demand curve shown earlier. a) Find the equilibrium point. Factors Affecting the Thai Natural Rubber Market Equilibrium: Demand and Supply Response Analysis Using Two-Stage Least Squares Approach Chadapa Chawananon Natural rubber is a major export crop and the sector is an important source of employment in Thailand. Using these curves as a base and model for perfectly competitive industries, the effects of shifts in demand and supply, the implementation of economic regulations, and changing market. We consider both short term and long term relationships, thanks to a lagged adjustment model and correct for the price endogeneity using a supply-demand equations system estimated with the 3 Stage. economics - Solutions manual 1. The concept of supply and demand is an economic model to represent these forces. Even though Joan is an economist, her knowledge of the market for jewelry boxes was based on experience and insight. 9 billion bushels and the annual demand is 2. The basic way to calculate this is to use a graph with both the supply and demand lines on it. For a certain commodity the supply equation is given by: S = 2p + 5 At a price of$1, there is a demand for 19 units of the commodity. The first part of figuring out demand is to find the marginal utility each good provides and the rate of substitution between the two goods—that is, how many units of x the consumer is willing. Find the Demand Equation The quantity demanded x each month of Russo Espresso Makers is 250 when the unit price p is $170; the quantity demanded each month is 500 when the unit price is$160. Questions related to quadratic equations and functions cover a wide range of business concepts including cost/revenue, break-even analysis, supply/demand, market equilibrium, and so on. Problems 7-10 deal with a decrease in demand and an increase in supply. Notice that the price actually paid by the consumer is only 7 - 2 or 5 and the supplier receives $7. Solving for the Equilibrium -To find the solution to the a system of equations with two variables, find a way to substitute one equation into the other. Open a new Excel spreadsheet and enter the data in a table as shown in this example. The relationship between supply and demand results in many decisions such as the price of an item and how many will be produced in order to allocate resources in the most cost-effective and efficient way. 3 in Finite Mathematics, Applied Calculusand Finite Mathematics and Applied Calculus) For best viewing, adjust the window width to at least the length of the line below. Let’s begin illustrating identification conditions using the normal supply and demand equations. In economics, Price Elasticity of Supply and Demand is the measurement of change in quantity of a service in accordance with the price change. B)the supply curve of a normal good shifts rightward. An increase in demand will create a shortage, which increases the equilibrium price and equilibrium quantity. Determine whether the ordered pair is a solution of the system of linear equations Solve a system using the elimination method Solve a system using the substitution method Solve a linear system by graphing Word Problems - Airplane Speed Word Problems - Supply and Demand Solving Multivariable Linear Systems. If the supply equation is linear, it will be of the form: P = a + b Qs. P(s 1) = P(d 1) Step 2. Also inverse demand curve formula. Use substitution to solve each system of equations. C)the demand curve for a normal good shifts rightward. The demand curve shows the amount of goods consumers are willing to buy at each market price. Aggregate demand is the overall demand for all goods and services in an economy. 80 per bushel, the annual supply for soybeans in the Midwest is 1. The labor supply and demand equations in Mexico and the US are. The laws of supply and demand help to determine what the market wants and how much. Note that it doesn't matter which one you use since the whole point is that they have to give you the same quantity. --You can edit this template and create your own diagram. P = 25 - 2 * Qd. Once your have the total, divide that by four (accounting for the next month). (a) Plot a scatter plot for the total market supply [4]. Market Demand. example for linear demand curve a) for linear demand, q = a bp,so bp=q bp= (a bp) b) note that = 1 when we are halfway down the demand curve c) see Figure 15. Identify the determinant of demand and. Let us suppose we have two simple supply and demand equations Qd = 20 - 2P Qs = -10 + 2P. c) Find the equilibrium price and quantity. Notice that this is an ifthen statement. 80 - Q = 20 + 2Q. At a price P * =$8, the supply of books is equal to demand: 24 buyers are willing to pay $8, and 24 sellers are willing to sell. Consider inserting a new equation to reflect this: Ps=Pd-2, and rearrange the equations for the supply and demand curves so that you you're solving for price, rather than quantity. (c) If the demand curve for beef is q = 140. We can find using the usual equation: This means. This is a supplemental video that shows my students how to graph supply and demand equations. Depicting a Free Trade Equilibrium: Large Country Case The adjoining graph depicts the supply and demand for wheat in the US market. a) Find the supply equation. Elasticity and Total Revenue/Total Expenditure 6. 80 per bushel, the annual supply for soybeans in the Midwest is 1. Note that a good way to check your answer for P is to substitute it back into both the supply and the demand equation and make sure you get the same answer. The demand for a good is elastic if a substitute for it is easy to find. To determine the point price elasticity of demand given P 0 is$1. By "demand for a commodity" at a given price is meant: "The total quantity of that commodity which buyers will take at different prices per unit of time". How to find equilibrium price with supply and demand equations, 4) Plug your equilibrium price into either your demand or supply. A multielement PU is also known as a "feedback region. Where money supply generally is an underpinning of economic activity, it also is the ultimate determinant of prices and inflation. Find the Demand Function Since the demand function is linear, we know that it can be written in slope-intercept form as p mx b We need to find the m and b from the information regarding price and quantity. To calculate equilibrium price and quantity mathematically, we can follow a 5- step process: (1) calculate supply function, (2) calculate demand. when the price falls to $1. I am sure that if you knew any economics words before enrolling in this course those two words were supply and demand.$\endgroup$– dismalscience Oct 14 '15 at 0:19. The price and quantity prevailing at market equilibrium point are known as equilibrium price and equilibrium quantity respectively. Supply, Demand, and Market Equilibrium Overview In this lesson, students will gain an understanding of how the forces of supply and demand influence prices in a market economy. This is a collection of diagrams for supply and demand. Sometimes the price per unit is a function x, say, p(x). Identifying High Quality Supply and Demand Levels By Joe Wright | May 20, 2014 Supply and demand levels can be found on any and all price charts, on every time frame for any instrument, but not all SD levels are created equal. Determinants of Price Elasticity 7. 8 billion bushels. If you need to produce a 'supply and demand' style chart using Excel, the following procedure for Excel 2013 and Excel 2010 could be useful:. Provide your answer in simplified form. 2; Solve the equations for the equilibrium price and quantity. On the one hand, we find that the time-series of demand-determined output fluctuations, that is the time-series of output constructed by putting all supply disturbance realiza- tions equal to zero, has peaks and troughs. Demand is an economic principle that describes a consumer's desire and willingness to pay a price for a specific good or service. In equilibrium,. 60 per bushel, the daily supply for wheat is 450 bushels and the daily demand is 645 bushels. Answer: The supply curve for the gasoline shifts from S 1 to S 2. Skip navigation Finding Equilibrium using Linear Demand and Supply Equations. Is this equilibrium stable? (Explain). The Short Run supply curve has two segments. To find the competitive equilibrium price and quantity in a market, we need to solve a pair of simultaneous equations—the demand curve , and the supply curve —for P and Q. ---- Answer: To find the equilibrium quantity, simply set both of these equations equal to each other. It focusses on the ancient laws of supply and demand and how price moves in a free-flowing market. The central bank controls the monetary base, expanding or contracting it at will, according to the needs of the economy. 25 for demand and 0. So your equation become 0=50-100P 100P=50 P=50/100 - - - > P=0. If the quantity supplied decreases by 4 at every price (so that the supply curve shifts to the left), the equilibrium price will change from: To solve for equilibrium price, set Qd equal to Qs. As another example an economist may look at the amount of money a. The formula will be F4 = (D1 + D2 +D3) ÷ 4. The demand curve shows the amount of goods consumers are willing to buy at each market price. 8 billion bushels. Supply and Demand is one of the core strategies used in trading. When the price increases to$5. Demand is given by P=150 - (1/6)Q and the supply given by P=50 + (1/3)Q a)Graph the functions and calculate the equilibrium price and quantity b) suppose a price is estabilished by the government in this market at a price of $120. The laws of supply and demand help to determine what the market wants and how much. Qd = 66-3P. 25Qs Demand curve = P =$85 - $0. How do you know if a good is a complement or a substitute from given demand and supply functions? [closed] You don't actually need the supply functions at all. But Excel only accepts x values first in the left most column, then however many dependent y variable columns one wants. The point where the demand and supply curve cross is called the equilibrium point $$(q^*, p^*)$$. (1) You can use the graphs and show that shifts the labor supply equation out (show in a graph that this implies ). Definition: Law of demand tells us that consumers will respond to a price drop by buying more, but it does not tell us how much more. What format do I use to obtain the equation of price of p=$20, which would sell x=42 sneakers each week. Algebra of the supply curve Since the demand curve shows a positive relation between quantity supplied and price, the graph of the equation representing it must slope upwards. Go through all the prices P = 1 through P = 10 in this fashion, and you'll have plotted out the demand curve. Show your solution graphically. Supply and demand are market forces that determine the price of a product. -Only when the supply and demand of labour are equal is there no involuntary unemployment: it is here that the market clears! IN A SLUMP:. 25 for demand and 0. There is currently no tax applied to this market. Mastering managerial economics involves calculating values, with the ultimate goal of determining how to maximize profit. 80 - Q = 20 + 2Q. We start by deriving the demand curve and describe the characteristics of demand. The convention is for the demand curve to be written as quantity demanded as a function of price. Questions related to quadratic equations and functions cover a wide range of business concepts including cost/revenue, break-even analysis, supply/demand, market equilibrium, and so on. If the supply equation is linear, it will be of the form: P = a + b Qs. Find the supply and demand equations, assuming they are linear. Identify the determinant of demand and. The demand equation for a certain type of printer is given by: D=-200p + 35,000 The supply equation is predicted to be: s= -p^2 + 400p - 20,000 Find the equilibrium price Do I solve for p first in the demand equation, and then input the answer into the supply equation? I did that and then would get: s = -175^2 + 400(175) -20,000 Am I doing this. What format do I use to obtain the equation of price of p=$20, which would sell x=42 sneakers each week. (A) Find. txt) or read online for free. (a) (10 points) Solve for the equations of demand and supply in this market and sketch the demand and supply curves. You can easily demonstrate that the labor supply curve has a positive slope by deriving one with your students. The market supply curve shows the combined quantity supplied of goods at different prices. Money, either in the form of currency or as bank reserves, is a liability of the central bank. Recall that in Chapter 1 of our book you learned linear regression, and supply and demand. Suppose that initially, the equations for demand and supply are Qd = 48 - 4P and Qs = 4P - 16, respectively. It focusses on the ancient laws of supply and demand and how price moves in a free-flowing market. To calculate a more exact measure of elasticity at a particular point on a supply or demand curve, we need to think about infinitesimally small changes in price and, as a result, incorporate mathematical derivatives into our elasticity formulas. Two ways to solve this. The safety stock needed to. In the same way, the equations in SEM can also be solved. Thru their actions in the market, the participants in the Forex market are constantly shifting the supply and demand of currency pairs, causing the price to fluctuate. In a perfect market the supply curve is Qs=9p and the demand curve is Qd=100-p. Mathematical example: Suppose P = 20 -. Dynamic supply and demand equations for particleboard using a threestage squares simulation (3SLS) were estimated. The supply and demand curves intersect at P* and Q*, which are the equilibrium price and quantity. The equilibrium price is determined by finding the point where both supply and demand are the same value, i. In addition to the demand schedule and the demand curve, the buyers' demand for a good can also be expressed a third way—algebraically, using a demand equation. The shortage can be calculated as follows. and supply shocks. When the quantity of money demanded increase, the price of money (interest rates) also increases, and causes the demand curve to increase and shift to the right. In other words, the monopolist chooses Q to maximize TR, and charges "as much as he can get away with"--the highest price consumers will pay for that profit-maximizing Q. A new video on Linear Supply equations. To get the base, find equilibrium quantity (Q*). 25q i dont just need this solved. The laws of supply and demand help to determine what the market wants and how much. Elasticity and tax revenue. 1 Self Assessment Solutions Linear Economic Models 1. "Price elasticity of supply measures how responsive producers are to a change in the price of good. where Q D is the quantity demanded of a good and P is the price of the good. Elasticity in the long run and short run. Supply and demand graph template to quickly visualize demand and supply curves. The guiding principle in the development of Matching Supply with Demand has been “real operations, real solutions. b) Sketch a graph. The model is not without its problems and shortcomings, many of which have been discussed in previous post: equilibrium assumption, excluding non-linearities, and a host of. To find the demand equation, we use the two price/quantity pairs and. To find the Demand Curve and Function for a single product, one must understand that in standard Economics, the relationship is inverse: Price Y determines Units Demanded X, or f(y) = x. Distribute a copy of Activity 4 to each student. Price elasticity of demand and price elasticity of supply. In this Leibniz, we show how to model the effects of a supply or demand shock mathematically. Since supply is much more than demand, the price is going. 4 in the P axis. Excess demand is the situation where the price is below its equilibrium price. the monetary policy rule of the central bank. The equilibrium price for dog treats is the point where the demand and supply curve intersect corresponds to a price of$2. The Short Run supply curve has two segments. 90 per bushel, the daily supply increases to 750 bushels and the daily demand decreases to 495 bushels. Is this equilibrium stable? (Explain). How do you know if a good is a complement or a substitute from given demand and supply functions? [closed] You don't actually need the supply functions at all. This module you will finally learn what all the fuss is about. Find the demand quantity and the supply quantity at a price of $30 each. variation of the simple supply-and-demand model taught in microeconomics, where the purpose is to determine equilibrium price and quantity in a market. Before you enter into an agreement, it is important to comprehend the conditions and conditions of the arrangement. The labor supply and demand equations in Mexico and the US are. Point of Equilibrium for Supply and Demand$$D(q_e)=S(q_e) Use MathJax to format equations. Elasticity 1. It is located at the intersection of the supply and the demand curve (i. He illustrates the new way in an example below: Assuming constant elasticity Demand: Q = AP ^ (-0. Think about the shift variables for demand, and the shift variables for supply. Demand is given by P=150 - (1/6)Q and the supply given by P=50 + (1/3)Q a)Graph the functions and calculate the equilibrium price and quantity b) suppose a price is estabilished by the government in this market at a price of$120. It was developed by Dennis Buchholz Im struggling to get it the settings are 3,85 port 76,14 please if you have it share it thank you. Algebra -> Quadratic Equations and Parabolas -> SOLUTION: The demand and supply equations for a certain item are given by D = –5p + 40 S = –p2 + 30p – 8 Find the equilibrium price. 2x + 9p - 93 = 0 and 3x - 14p + 108 = 0 a. You should reflect on the moral of the problems: When both demand decreases and supply increases, we can be sure that price will decrease. and supply shocks. 3 on production schedules for boats and leases for airplanes). Notice that the quantity supplied does not equal the quantity demanded when P = 4. This handout explains concepts and provides. Textbook solution for Calculus of a Single Variable 11th Edition Ron Larson Chapter 10 Problem 28RE. QUADRATIC EQUATIONS AND FUNCTIONS Quadratic equations and functions are very important in Business Math. The inverse demand function is useful in deriving the total and marginal revenue functions. However, for the demand, as the price goes up, the number of items goes down. Denis Sidorov1,3, , Ildar Muftahov1,2, Nikita Tomin1, , Dmitriy Karamov1, Daniil Panasetsky1, , Aliona Dreglea1,3, Fang Liu4, and Aoife Foley5, Manuscript received. 10 – (1/500)Q = 2 + (1/500)Q Solving, we find Q = 2000. 3) Supply: Q = BP ^ (0. This is a collection of diagrams for supply and demand. Calculate the profit at each level, looking for the maximum profit and answer the questions at that point. Supply and demand is one of the basic principles of economics and the free market. Market Demand and Supply - Equilibrium Last summer, the number of tourists has fallen sharply in Syria, as a result of the political unrest over last year. Find the market equilibrium. So the price elasticity of demand for soft drinks equals. The quantity supplied is lower than the quantity demanded by the consumers. 1 THE PRICE ELASTICITY OF DEMAND Three main factors influence the ability to find a substitute for a good: Luxury Versus Necessity • A necessity has poor substitutes, so the demand for. 5Q P = 22 + 3Q. For the pair of supply-and-demand equations, where x represents the quantity demanded in units of 1,000 and p is the unit price in dollars, find the equilibrium quantity and the equilibrium price. NOTE: We know that both supply and demand functions are linear. Firstly, let's look at what the inverse demand and supply equations are actually representing. Once we have the equilibrium quantity, we can find the equilibrium price by plugging into either the supply equation or the demand equation. From trade association data you are able to obtain estimates for the own price elasticities ofdemand and supply on the world markets as ?0. when the price falls to $1. demand and supply on the world markets as −0. A linear supply curve can be plotted using a simple equation P = a + bS. A) Find the Supply equation B) Find the Demand equation -End Question I'm stumped, thanks in advance for the help. Demand is given by P=150 - (1/6)Q and the supply given by P=50 + (1/3)Q a)Graph the functions and calculate the equilibrium price and quantity b) suppose a price is estabilished by the government in this market at a price of$120. Recall all demand curves have negative slope. For example, if the demand function for USA is: P = 14 + 0. The following chart illustrates the excess demand and excess supply. In the pair of supply and demand equations below, x represents the quantity demanded in units of a thousand and p the unit price in dollars. How the step graph for a small market becomes a smooth curve for a larger market. 49 per bushel, the daily supply for wheat is 305 bushels, and the daily demand it 515 bushels. P = 25 - 2 * Qd. How to solve for equilibrium price and quantity There are some minor verbal mistakes, which I. We consider both short term and long term relationships, thanks to a lagged adjustment model and correct for the price endogeneity using a supply-demand equations system estimated with the 3 Stage. The actual stock of money is determined by the point where the supply and demand curves intersect. ) or service (transportation, health care, education etc. When the price is raised to $1. The demand equation can be written as a regular linear equation and an inverse equation. Qd = a - b(P) Q = quantity demand; a = all factors affecting price other than price (e. We'll also learn what the "price-intercept" is, its significance and how it can easily be determined using the demand equation. What format do I use to obtain the equation of price of p=$20, which would sell x=42 sneakers each week. The future demand is intrinsically coupled to future supply chain decisions. If price is above the market equilibrium, then there is a surplus. Solve systems of equations by graphing A system of linear equations contains two or more equations e. Demand and supply curves can be represented with equations. These prices are set using equations that determine how many items to make and whether to raise or lower prices to keep that demand constant. a = plots the starting point of the supply curve on the Y-axis intercept. A change in demand (or supply) shifts the whole curve. ) o Single-equation estimation involves estimating only the one equation of interest, but we still need to consider the variables that are in the other equation(s). This allows us to create what we call two ordered pairs (x 1,y 1) and (x 2, y 2). Find the demand quantity and the supply quantity at a price of \$30 each. C)the demand curve for a normal good shifts rightward. Meaning of Price Elasticity of Demand 3. Find the equation of the line for the demand equation. Generally you need to solve the functions for quantity (Q) and change the intercept. Keywords: gold market, simultaneous equations method, two stage least squares, demand-supply analysis. Estimated Supply And Demand Equations Table. d) Graph the two equations in the same coordinate syste. So your equation become 0=50-100P 100P=50 P=50/100 - - - > P=0. pdf), Text File (. 29, which differs from the common assumption of a perfectly inelastic short-run supply curve. 20 per bushel,the daily supply decreases to 350 bushels,and the daily demand increases to 980 bushels. Use a system of linear equations to find the equilibrium point for supply and demand problems. Impact on Demand. Arc elasticity of demand (arc PED) is the value of PED over a range of prices, and can be calculated using the standard formula: More formally, we can say that PED is the ratio of the quantity demanded to the percentage change in price. The demand curve is a graph used in economics to demonstrate the relationship between the price of a product and the demand for that same product. Supply and Demand of Loanable Funds (With Explanations)! Subject Matter: To improve upon the classical macro theory by taking the influence of money into account, a school of thought developed which is popularly called the neoclassical school. Potential benefits of installing a demand recirculating system during your next new construction or home retrofitting project, include: Demand recirulating pumps have the potential to solve the problem of a long. b) Find the demand equation. Supply Chain Optimization Problem Examples. Determinants of Price Elasticity 7. Learn vocabulary, terms, and more with flashcards, games, and other study tools. We will revisit finding the maximum and/or minimum function value and we will define the marginal cost function, the average cost, the revenue function, the marginal revenue function and the marginal profit function. Consider a demand curve of the form. a) Find the supply equation. This is the currently selected item. demand, or MD c = 0. And there is a price, the wage rate, which should clear the market. These prices are set using equations that determine how many items to make and whether to raise or lower prices to keep that demand constant. Get an answer for 'The demand curve for labor is Q=100-5P. Math 103 Section 1.
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2019-12-10 02:27:39
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https://www.aimsciences.org/article/doi/10.3934/dcds.2001.7.85
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# American Institute of Mathematical Sciences
January 2001, 7(1): 85-90. doi: 10.3934/dcds.2001.7.85
## On Chenciner-Montgomery's orbit in the three-body problem
1 School of Mathematics, University of Minnesota, Minneapolis, MN 55455, United States
Received June 2000 Revised September 2000 Published November 2000
Recently A. Chenciner and R. Montgomery found a remarkable periodic orbit for a three-body problem by variational methods. On this orbit all masses chase each other along a figure-eight circuit without any collision, and the solution curve is indeed a minimizer of the action functional on a properly chosen path space. One technical difficulty, where numerical integration had been used in their proof, is to show that the minimizing orbit does not experience any collision. In this paper a short analytical proof will be presented.
Citation: Kuo-Chang Chen. On Chenciner-Montgomery's orbit in the three-body problem. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 85-90. doi: 10.3934/dcds.2001.7.85
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2019 Impact Factor: 1.338
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2020-11-27 14:47:43
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http://pjbull.github.io/civil_conflict/simulation.html
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## Modeling and Simulating Political Violence and Optimizing Aid Distribution in Uganda
### Simulating events in space and time
#### Events in space
The violence and civil conflict events described in the ACLED dataset are coded with both a latitude and a longitude. One assumption we make is that these events are distributed due to underlying causes such as population centers, road access, historical land ownership, and other features that make conflicts more or less likely. It is difficult to directly create a generative model for these probabilities, so we will make the further assumption that the distribution of the data already incorporates and is representative of these factors. In effect, we treat the entire country of Uganda as a probability distribution from which geospatial conflict events could be sampled. We took historical conflict location data from the entire ACLED data set and smoothed it using a Matérn covariance function. Figure 2 shows this smoothing applied to the same conflicts depicted in figure 1.
Figure 2
We used this smooth function as a kernel-density esitmate (KDE). This estimate (i.e., the empirical distribution of the conflict data), has a complex functional form which makes it challenging to sample from. However, for any given coordinate it is quite simple to get the probability of an event. Given this property of our KDE, we can apply Monte Carlo sampling techniques to generate samples from this probability distribution. Visualizing the distribution, we can see that it is multi-modal with regions of low density between the modes. Because of these properties, we opted to use slice sampling to generate draws from the distribution.
Figure 3 shows the first 1,000 samples from this probability distribution. (Note: we throw away samples that occur over water or outside of country boundaries.) Figure 4 shows the distribution of the samples as a two-dimensional histogram.
Figure 3
Figure 4
When slice sampling, there are a number of parameter choices that are imporant. We want to choose our rectangle widths, burn-in, and thinning parameters appropriately. In testing, a thinning value of 10 reduced autocorrelation to less than 0.1 at a time lag of 1. We can see this result in figure 5. We used the Gelman-Rubin potential scale reduction factor to determine if we were observing favorable mixing. Generally, a value less than 1.1 indicates good mixing. In both of our dimensions, the Gelman-Rubin statistic was less than this threshold. We also calculated the Geweke statistic, which is used to indicate convergence. A value less than 2 indicates convergence and for our draws, this statistic was $\ll$ 1. We can also examine convergence by looking at the trace plots for the samples. As we see in figure 6 these appear stationary.
Figure 5
Figure 6
#### Events in time
As a modeling assumption, we separate the dimensions of space and time as being independent. To model events in time across the country, we use an autoregressive Poisson GLM. While standard autoregressive models create a linear relation between a future value and a previous value, the Poisson GLM permits a linear relation between previous data and the mean of a Poisson distribution. This will allow us to retain the probabilistic interpretation of the events in time.
In order to model events using a Poisson distribution, we must discretize our time dimension. We opted for month-long increments. The thinking behind this decision is that we want to use sample draws to run our aid optimizations. If a model like this were to be used in planning for future conflicts, having a month-long window for a plan seems like a good balance between precision and logistic concerns.
The Autoregressive Poisson GLM model can be described as a log-linear relationship between the number of events of political violence and the mean of a Poisson distribution. We start with a timeseries, $\mathbf{X} = \{x_0, x_1,\ldots, x_N\}$, of $N$ counts of events at each discretized point in time. We also start with a lag $\Lambda$ that is the number of previous time steps to include in the model. We can now describe our features at time $t$ as the $\Lambda$ previous time steps: $\mathbf{X_{t, \Lambda}} = \{x_{t-\Lambda}, x_{t-\Lambda-1},\ldots, x_t\}$.
The linear predictor in the autoregressive model at a time step is $\eta_t$, and it is related to the mean of the Poisson distribution, $\mu_t$, by its canonical link function, $\log$.
\begin{aligned} \eta_t &= \mathbf{X}_{t, \Lambda}'\beta. \\ \mu_t &= \log(\eta_t) = \log(\mathbf{X}_{t, \Lambda}'\beta) \\\end{aligned}
Finally, the only parameter to the Poisson distribution is this mean, so the distribution of counts, $k$, at some time t+1 can be given by:
\begin{aligned} p(k | \mu_t) &= \frac{\mu_t^k}{k!}e^{-\mu_t} \\ p(k | \mathbf{X}_{t, \Lambda}, \beta) &= \frac{\log(\mathbf{X}_{t, \Lambda}'\beta)^k}{k!}e^{-\log(\mathbf{X}_{t, \Lambda}'\beta)} \end{aligned}
We can fit this model by using Fisher scoring to calculate $\beta_\mathrm{MLE}$, the coefficients of the model. Figure 7 shows this model as compared to actual rates of conflict incidence.
Figure 7
#### Putting together the spatial and temporal
We can now use our draws over space and time to generate a simulation of future conflicts in Uganda. These simulated scenarios will serve as the basis of our aid delivery optimization. The combination of modeling conflict events in space and time along with optimizing aid delivery could prove helpful to organizations such as the Red Cross in ordering supplies, allocating staff and volunteers, and developing infrastructure.
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2017-09-23 03:44:14
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https://docs.pumas.ai/latest/
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# Pumas
Pumas (PharmaceUtical Modeling And Simulation) is a suite of tools to perform quantitative analytics of various kinds across the horizontal of pharmaceutical drug development. The purpose of this framework is to bring efficient implementations of all aspects of the analytics in this domain under one cohesive package. Pumas 1.0 currently includes:
• Non-compartmental Analysis
• Specification of Nonlinear Mixed Effects (NLME) Models
• Simulation of NLME model using differential equations or analytical solutions
• Deep control over the differential equation solvers for high efficiency
• Estimation of NLME parameters via Maximum Likelihood and Bayesian methods
• Parallelization capabilities for both simulation and estimation
• Mixed analytical and numerical problems
• Interfacing with global optimizers for more accurate parameter estimates
• Simulation and estimation diagnostics for model post-processing
• Global and local sensitivity analysis routines for multi-scale models
• Bioequivalence analysis
Additional features are under development, with the central goal being a complete clinical trial simulation engine which combines efficiency with a standardized workflow, consistent nomenclature, and automated report generation. All of this takes place in the high level interactive Julia programming language and integrates with the other packages in the Julia ecosystem for a robust user experience.
Pumas is covered by the Julia Computing EULA. Pumas is a proprietary product developed by Pumas-AI, Inc. It is available free of cost for educational and research institutes. For commercial use, please contact [email protected]
## Getting Started: Installation and First Steps
One can start using Pumas by invoking it from the REPL as below.
using Pumas
To start understanding the package in more detail, please checkout the tutorials at the start of this manual. We highly suggest that all new users start with the Introduction to Pumas tutorial! If you find any example where there seems to be an error, please open an issue.
If you have questions about usage, please join the official Pumas Discourse and take part in the discussion there. There is also a #pumas channel on the JuliaLang Slack for more informal discussions around Pumas.jl usage.
Below is an annotated table of contents with summaries to help guide you to the appropriate page. The materials shown here are links to the same materials in the sidebar. Additionally, you may use the search bar provided on the left to directly find the manual pages with the appropriate terms.
### Tutorials
These tutorials give an "example first" approach to learning Pumas and establish the standardized nomenclature for the package. Additionally, ways of interfacing with the rest of the Julia ecosystem for visualization and statistics are demonstrated. Thus we highly recommend new users check out these tutorials before continuing into the manual. More tutorials can be found at https://tutorials.pumas.ai/
### Basics
The basics are the core principles of using Pumas. An overview introduces the user to the basic design tenants, and manual pages proceed to give details on the central functions and types used throughout Pumas.
### Model Components
This section of the documentation goes into more detail on the model components, specifying the possible domain types, dosage control parameters (DCP), and the various differential equation types for specifying problems with analytical solutions and alternative differential equations such as delay differential equations (DDEs), stochastic differential equations (SDEs), etc.
### Analysis
This section of the documentation defines the analysis tooling. Essential tools such as diagnostics, plotting, report generation, and sensitivity analysis are discussed in detail in this portion.
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2020-10-31 13:24:25
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https://groupprops.subwiki.org/wiki/Transfer-closed_characteristicity_is_transitive
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# Transfer-closed characteristicity is transitive
This article gives the statement, and possibly proof, of a subgroup property (i.e., transfer-closed characteristic subgroup) satisfying a subgroup metaproperty (i.e., transitive subgroup property)
View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about transfer-closed characteristic subgroup |Get facts that use property satisfaction of transfer-closed characteristic subgroup | Get facts that use property satisfaction of transfer-closed characteristic subgroup|Get more facts about transitive subgroup property
## Definition
Suppose $H \le K \le G$ are groups such that $K$ is a transfer-closed characteristic subgroup of $G$ and $H$ is a transfer-closed characteristic subgroup of $K$. Then, $H$ is a transfer-closed characteristic subgroup of $G$.
## Proof
### Proof using given facts
The proof follows from facts (1) and (2).
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2020-12-02 18:45:24
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https://dsp.stackexchange.com/questions/linked/741?sort=hot
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5k views
### What happens when N increases in N-point DFT [duplicate]
I am curious about DFT, and I wrote a simple MATLAB code to test what happens when $N$ increases. I took a rectangular signal with length $L=15$, an then found th DFT of 16, 32 and 64 points. I looked ...
567 views
### What does it mean when you zero pad data and why does one do it? [duplicate]
I am struggling to understand what effect zero padding data would have on EMG data. How does it improve the signals? Any help in simple terms that's not too mathematical would be greatly appreciated!
256 views
100k views
### How do I implement cross-correlation to prove two audio files are similar?
I have to do cross correlation of two audio file to prove they are similar. I have taken the FFT of the two audio files and have their power spectrum values in separate arrays. How should I proceed ...
19k views
### Why images need to be padded before filtering in frequency domain
In image processing books , we are told that Images need to be padded while doing filtering in the frequency domain. Why we need that zero padding ?
5k views
whats is the effect of pading zeroes to a sequence in FFT? for eg: x[n]=[2 3 4 5] corresponds to X[K]=[14 -2+2i -2 -2-2i] while x[n]=[2 3 4 5 0 0 0 0] corresponds to X[K]=[14 0.58-9.65i -2+2i -3.414+...
5k views
### Frequency-domain zero padding - special treatment of X[N/2]
Suppose we wish to interpolate a periodic signal with an even number of samples (e.g. N=8) by zero-padding in the frequency domain. Let the DFT X=[A,B,C,D,E,F,G,H] ...
5k views
### Zero-pad before or after windowing for FFT
What's the correct way. Should I zero-pad a signal before or after applying a windowing function?
9k views
I would like to know what are the advantages/disadvantage of zero padding with respect to frequency measurement and amplitude measurement
8k views
### FFT of size not a power of 2
My question is regarding the input size of a signal which is not a power of 2 and we have to take the fft of it. Some solutions say that suppose if we want to take the fft of 1800 we should zero pad ...
3k views
### disadvantages of FFT, it can not extract enough frequencies without enough samples
Let's say sampling rate is $Fs = 44\mathtt{kHz}$, now I have $N = 2048$ samples, then I can get $N/2 + 1 = 1025$ frequencies. I'm confused by Matlab's FFT documentation that says the frequencies are ...
1k views
### The frequency response function (FRF) fails to detect the antiresonance of a system
I am trying to identify a vibrational systems by computing the frequency response function (FRF) of the system when a chirp signal is applied to its input. After comparing the FRF computed and the ...
848 views
### Mathematical justification for zero padding?
This question asks what's the point of zero padding. The accepted answer is certainly very insightful, but I don't understand a big chunk of it: Zero padding allows one to use a longer FFT, which ...
For a given signal, I've been told that you can pad the vector with $0$s at the end to get a larger DFT, and as a result get more precision in frequency bins. What are the limits to this approach? ...
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2020-08-14 05:43:29
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https://mathhelpboards.com/threads/prove-the-summation-identity.4481/
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# TrigonometryProve the summation identity.
#### anemone
##### MHB POTW Director
Staff member
Hi,
I have been trying to solve this difficult problem for some time and I thought of at least two ways to prove it but to no avail...the second method that I thought of was to employ binomial expansion on the denominator and that did lead me to the result where it only has x terms in my final proof, but it did not lead to the desired result (and I should be able to tell beforehand that I wouldn't get anywhere near to the desired form of the result by expanding the expression on the denominator too)...I am really at my wit's end and really very mad at myself now and I'd be grateful if someone could throw some light on to the problem...thanks in advance.
Let $$\displaystyle -1<x<1$$, show that $$\displaystyle \sum_{k=0}^{6} {\frac{1-x^2}{1-2x\cos (\frac{2\pi k}{7})+x^2}}=\frac{7\left(1+x^7 \right)}{1-x^7}$$.
#### Opalg
##### MHB Oldtimer
Staff member
Let $$\displaystyle -1<x<1$$, show that $$\displaystyle \sum_{k=0}^{6} {\frac{1-x^2}{1-2x\cos (\frac{2\pi k}{7})+x^2}}=\frac{7\left(1+x^7 \right)}{1-x^7}$$.
This problem cries out for the use of complex numbers. I'll prove a slightly more general result.
Let $\omega = e^{2\pi i/n}$, with complex conjugate $\overline{\omega} = \omega^{-1}$. The $n$th roots of unity are $\omega^k\ (0\leqslant k\leqslant n-1)$, and $\displaystyle 1-x^n = \prod_{k=0}^{n-1}(1-\omega^k x)$. It follows that there must be a partial-fraction decomposition of the form $$\frac n{1-x^n} = \sum_{k=0}^{n-1}\,\frac{s_k}{1-\omega^k x}.$$ To find the coefficients $s_j$, multiply both sides by $1-\omega^jx$ to get $$\frac {n(1-\omega^jx)}{1-x^n} = s_j + (1-\omega^jx)f(x)$$ for some function $f(x)$ that is continuous at $\omega^{-j}.$ Then $$s_j = \lim_{x\to\omega^{-j}}(s_j + (1-\omega^jx)f(x)) = \lim_{x\to\omega^{-j}}\frac {n(1-\omega^jx)}{1-x^n} = \lim_{x\to\omega^{-j}}\frac {-n\omega^j}{-nx^{n-1}} = 1,$$ (using l'Hôpital's rule to evaluate the limit). Therefore $$\frac n{1-x^n} = \sum_{k=0}^{n-1}\,\frac1{1-\omega^k x}.$$ Multiply that by 2, and use the facts that $\overline{\omega}^k = \omega^{n-k}$ and $\omega^k + \overline{\omega}^k = 2\cos\bigl(\frac{2k\pi}n\bigr)$, to get $$\frac {2n}{1-x^n} = \sum_{k=0}^{n-1}\,\biggl(\frac1{1-\omega^k x} + \frac1{1-\overline{\omega}^k x}\biggr) = \sum_{k=0}^{n-1}\,\frac{2-2x\cos\bigl(\frac{2k\pi}n\bigr)}{1-2x\cos\bigl(\frac{2k\pi}n\bigr) + x^2}.$$ Finally, subtract $n$ from both sides to get $$\frac{n(1+x^n)}{1-x^n} = \frac {2n}{1-x^n} - n = \sum_{k=0}^{n-1}\,\biggl(\frac{2-2x\cos\bigl(\frac{2k\pi}n\bigr)}{1-2x\cos\bigl(\frac{2k\pi}n\bigr) + x^2} - 1\biggr) = \sum_{k=0}^{n-1}\,\frac{1-x^2}{1-2x\cos\bigl(\frac{2k\pi}n\bigr) + x^2}.$$
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2021-06-23 05:49:08
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https://www.physicsforums.com/threads/find-direction-of-tangent-line-to-equation.572056/
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# Find direction of tangent line to equation
## Homework Statement
Find the direction of the line tangent to the curve $x^4+y^4=32$ at the point $(2, -2)$
## Homework Equations
Anything goes, we're in vector calculus now.
## The Attempt at a Solution
So, to find the tangent line, I was thinking of taking the gradient, but I'm not sure how to do that since the curve is in a squirrely format. I think they call that "implicit".
Perhaps the gradient is $4x^3i+4y^3j$?
and then at the given point we would have $32i-32j$
Is this actually the direction of the line tangent to the curve at the given point? Is this actually the answer? I don't really know.
SammyS
Staff Emeritus
Homework Helper
Gold Member
## Homework Statement
Find the direction of the line tangent to the curve $x^4+y^4=32$ at the point $(2, -2)$
## Homework Equations
Anything goes, we're in vector calculus now.
## The Attempt at a Solution
So, to find the tangent line, I was thinking of taking the gradient, but I'm not sure how to do that since the curve is in a squirrely format. I think they call that "implicit".
Perhaps the gradient is $4x^3i+4y^3j$?
and then at the given point we would have $32i-32j$
Is this actually the direction of the line tangent to the curve at the given point? Is this actually the answer? I don't really know.
No. $32\hat{i}-32\hat{j},,$ is the direction of greatest slope. The tangent line is perpendicular to this.
darn. so how do I find the perpendicular to that?
SammyS
Staff Emeritus
Homework Helper
Gold Member
darn. so how do I find the perpendicular to that?
Rotate the vector 90°.
jedishrfu
Mentor
Remember doing lines perpendicular to a line how did you figure out the slope?
Negative inverse right?
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2021-09-22 05:37:02
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https://zbmath.org/?q=an:0882.54005
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# zbMATH — the first resource for mathematics
Arcwise connectedness of the complement in a hyperspace. (English) Zbl 0882.54005
Let $$X$$ be a (metrizable) continuum and $$C(X)$$ be its hyperspace of non-degenerate subcontinua. It is known that $$C(X)$$ is arcwise connected. In this interesting paper the author shows that if $$\mathcal Y$$ is a countable closed subset of $$C(X)$$ and $$C(X)\smallsetminus \{Y\}$$ is arcwise connected for every $$Y\in {\mathcal Y}$$ then $$C(X)\smallsetminus {\mathcal Y}$$ is arcwise connected. This answers a question of S. B. Nadler jun. [Hyperspaces of sets, Monogr. Textb. Pure Appl. Math. 49 (1978; Zbl 0432.54007)].
##### MSC:
54B20 Hyperspaces in general topology 54D05 Connected and locally connected spaces (general aspects)
##### Keywords:
arcwise connectivity
Full Text:
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2021-09-16 18:25:14
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http://fun-phys.blogspot.kr/2016/06/standing-waves-in-open-end-air-columns.html
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# Standing Waves in Open End Air Columns
#### Quiz
Q) A pipe open at both ends resonates at a fundamental frequency $\ f_{open}.$ When one end is covered and the pipe is again made to resonate, the fundamental frequency is $\ f_{closed}.$ Which of the following expressions describes how these two resonant requencies compare?
A. $\ f_{closed}=f_{open}$ B. $\ f_{closed}=1/2f_{open}$
C. $\ f_{closed}=2f_{open}$ D. $\ f_{closed}=3/2f_{open}$
Q2) Balboa Park in San Diego has an outdoor organ. When the air temperature increases, the fundamental frequency of one of the organ pipes
A. stays the same, B. goes down,
C. goes up, or D. is impossible to determine.
The sound emitted by the organ pipes increases in frequency as they warms up because the speed of sound increases in the increasingly warmer air inside them.
#### Standing Waves in Open End Air Columns Simulation
When the next simulation is not visible, please refer to the following link.
Sound Waves
As sound waves travel through the air in the pipe, they are reflected at each end and travel back through the pipe. (The reflection occurs even if an end is open, but the reflection is not as complete as when the end is closed.) If the wavelength of the sound waves is suitably matched to the length of the pipe, the superposition of waves traveling in opposite directions through the pipe sets up a standing wave pattern. The wavelength required of the sound waves for such a match is one that corresponds to a resonant frequency of the pipe. The advantage of such a standing wave is that the air in the pipe oscillates with a large, sustained amplitude, emitting at any open end a sound wave that has the same frequency as the oscillations in the pipe. This emission of sound is of obvious importance to, say, an organist.
Standing Waves in Two Open Ends
The simplest standing wave pattern that can be set up in a pipe with two open ends is shown in Fig. 1a. There is an antinode across each open end, as required.
Figure 1. (a) The simplest standing wave pattern of displacement for (longitudinal) sound waves in a pipe with both ends open has an antinode (A) across each end and a node (N) across the middle. (The longitudinal displacements represented by the double arrows are greatly exaggerated.) (b) The corresponding standing wave pattern for (transverse) string waves.
There is also a node across the middle of the pipe. An easier way of representing this standing longitudinal sound wave is shown in Fig. 1b—by drawing it as a standing transverse string wave. The standing wave pattern of Fig. 1a is called the fundamental mode or first harmonic.
Figure 2. Motion of elements of air in standing longitudinal waves in a pipe, along with schematic representations of the waves. In the schematic representations, the structure at the left end has the purpose of exciting the air column into a normal mode. The hole in the upper edge of the column ensures that the left end acts as an open end. The graphs represent the displacement amplitudes, not the pressure amplitudes. In a pipe open at both ends, the harmonic series created consists of all integer multiples of the fundamental frequency: $\ f_{1}, 2f_{1}, 3f_{1}$, . . . .
For it to be set up, the sound waves in a pipe of length L must have a wavelength given by $\ L = \lambda /2$, so that $\lambda = 2L$. Several more standing sound wave patterns for a pipe with two open ends are shown in Fig. 2 using string wave representations.The second harmonic requires sound waves of wavelength $\lambda = L$, the third harmonic requires wavelength $\lambda = 2L/3$, and so on.
More generally, the resonant frequencies for a pipe of length $\ L$ with two open ends correspond to the wavelengths
$\lambda = \frac{2L}{n}, ~~~~~for~~ n=1,2,3, . . .$
where $\ n$ is called the harmonic number. Letting $\ v$ be the speed of sound, we write the resonant frequencies for a pipe with two open ends as
$f=\frac{v}{\lambda }=\frac{nv}{2L}, ~~~~~for~~ n=1,2,3, . . .$
It is interesting to investigate what happens to the frequencies of instruments based on air columns and strings during a concert as the temperature rises. The sound emitted by a flute, for example, becomes sharp(increases in frequency) as the flute warms up because the speed of sound increases in the increasingly warmer air inside the flute. The sound produced by a violin becomes flat(decreases in frequency) as the strings thermally expand because the expansion causes their tension to decrease.
Musical instruments based on air columns are generally excited by resonance. The air column is presented with a sound wave that is rich in many frequencies. The air column then responds with a large-amplitude oscillation to the frequencies that match the quantized frequencies in its set of harmonics. In many woodwind instruments, the initial rich sound is provided by a vibrating reed. In brass instruments, this excitation is provided by the sound coming from the vibration of the player’s lips. In a flute, the initial excitation comes from blowing over an edge at the mouthpiece of the instrument in a manner similar to blowing across the opening of a bottle with a narrow neck. The sound of the air rushing across the edge has many frequencies, including one that sets the air cavity in the bottle into resonance.
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2017-05-23 16:46:21
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http://www.gmhydro.it/it/en/bearings/
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THRUST AND GUIDE BEARINGS
General Information
The turbine-generator unit presents remarkable masses under rotation and considerable inertia forces, that increase if masses are not balanced, come into play. The runner may be subject to lateral forces due to the asymmetric flow that strikes it and the generator may be subject to magnetic pull. These forces associated to the weight of the rotating parts must be controlled by means of the BEARINGS. In the case of HORIZONTAL machines, the main loads on the bearings are due to the weights that act in perpendicular with the machine shaft and the axial loads, also present, are smaller than those in the corresponding vertical unit because in this case there is no weight along the shaft. Making reference to a vertical unit, the axial loads are discharged by the THRUST BEARING and the radial ones by the GUIDE BEARINGS. The number of guide bearings depends on the type of turbine-generator unit and essentially by the length of the shaft line. A verification if the number adopted is correct is carried out by means of a calculation of the flexural critical speeds. The bearings may be oil, grease or water lubricated and, depending on the application, we can find sliding smooth (the most used) or rolling (rollers, balls) surfaces.
GUIDE BEARINGS
In HORIZONTAL UNITS, the definition of guide maybe improper because they counteract also the weight of rotating parts. Anyway, once made this clarification, we can say that, depending on their behavior, in this type of machines we can identify two main types of bearings: STIFF and TILTING. The stiff ones are used if there is no appreciable deformation of the shaft line, while the tilting ones may self-align making use of spherical seats (fig. 1). Very often they are combined with the thrust bearing. The (fig. 2) illustrates one combined bearing with two opposite counterthrusts that prevent the axial sliding in both directions. Depending on the specific pressure on active parts and on the rotation speed of the shaft, the lubrication may be forced, by the aid of pumps, or inside the tank itself by using rings placed on the shaft that are dragged in rotation by the shaft itself. Their use is limited in first approximation to the following values of the peripheral speed U of the shaft: 2,5 m/s <U< 13 m/s out of this interval it is necessary to carry out the forced lubrication. Indicating by D the shaft diameter, the length of the active part of the bearing is on average L= (0,8 ÷ 1)
In VERTICAL UNITS, the main function is to keep the shaft aligned vertically and support any radial force that can take place during the operation. The tilting guide pads (fig. 3.1 and 3.2) are usually adopted, even if we can find bearing with bushings in case of mean-small units. The lubrication may be natural with a thermal exchange coil placed inside the tank or forced by the aid of pumps and external exchangers. In general the use of this type of bearings requires the presence of a manifold, integral or placed on the shaft and, according to the dimension of the machine the number of pads may vary from 6, 8, 10, 12, .... The lining materials that can be used of the bearings are very different: rubber, bronze, resin, white metal, the last one is the most used one. An example may be SnSb8Cu7 UNI 4515 that presents in room temperature a hardness BRINNEL of 23÷27 MPa.
Fig 1
Fig 2
Fig 3.1
Fig 3.2
THRUST BEARINGS
In general, in vertical machines, the thrust bearings are positioned above the generator and often they are combined with one guide bearing (fig. 4). Very important is the resting point of each pad with respect to the load barycenter. In monodirectional machines, the position of the resting point is not barycentric and this favors the formation of an oil film between pad and thrust runner. In reversible turbines (pump-turbines) the resting point must be located halfway of the pad length and therefore this will cause a more critical situation with respect to the monodirectional machines. As a general dimensioning criterion, we keep a specific pressure on the pad surface of 2,5÷4 MPa, (areaching even 6 MPa) depending on the material used. For what concerns the lubrication and the cooling system, it is valid what already said for vertical guide bearings. Generally, the maximum temperature measured on the pads does not exceed 80°C and that of the oil bath in which they are submerged 50÷60 °C. We can summarize 3 main aspects for a good operation of a thrust bearing:
1) limited specific pressure on the pad
2) homogeneous and generous cooling
3)surfaces in relative motion (pad and thrust runner) machined with narrow tolerances
Fig 4
Fig 4bis
Figure (fig 4 bis) represents the section of recent combined bearing installed in a plant of medium power.
The (Fig. 5) shows the construction drawing of a typical thrust pada
Fig 5
OIL INJECTION
The oil injection is more and more used between pad and thrust runner. The purpose is to separate the surfaces in touch in order to prevent the sliding at the moment of start up. This happens more easily after many days of stop of the machine because the surfaces in touch have the tendency to paste. The injection pressure must be such to generate a load that exceeds the rotating parts and causing in this way the detachment between thrust pads and rotating parts. All pads, or only some of them, may be equipped with a pocket for the injection (fig. 6, 7). Obviously the dimension of the pocket depends on the dimensions of the pad and of the load to lift up and must not be an obstacle for the formation of the oil film. Just for reference and in first approximation, the oil pressure could be of about 120 bar. The injection takes place before the turbine begins to move and it is a normal practice to disable it when the rotations reach about 90% of rated speed. During the stop phase, it is also usual to introduce the injection at about 90% of rated speed and to exclude it when the machine is completely stopped.
If we consider a circular pad, the diagram of (fig. 7) shows how the oil acts between the two surfaces and the values of discharge and pressure of the pump.
Fig 6
Fig 7
Fig 8
HYDROSTATIC BEARINGS
In general hydrostatic bearings are used mostly when the relative velocity between fixed and movable parts is low, as for example in telescopes. High relative speeds imply more entrainment of the oil or of any fluid passing through the fixed and the rotating parts, increasing in this way the power employed. Here below there are some considerations of general nature, made without entering into details that would render too heavy the subject matter, and anyway remaining inside the area concerning the applications for hydroelectric units. Differently from the traditional hydrodynamic bearings, where the oil film is granted by the relative speed between pad and rotating part, in hydrostatic bearings the oil film between the surfaces in relative motion is assured by a source under pressure, usually pumps. Generally, this lubrication procedure gives the following advantages:
• very low frictions
• separation of the surfaces to be lubricated made in safety
• high stiffness of the oil film, often higher than that in the mechanical structures
The negative aspect is connected to the fact that the set up is more complex, as it can be deduced in the following description. The oil supply can be made under constant pressure or at constant flow. This means that in the first case the load variations on the bearing will cause variations of the oil film and then of the oil flow through the pad, vice versa in the second case the variable becomes the pressure because the flow is constant, that is to say that film variations due to load variations will imply a variation of the pressure generated by the feeding pump. We refer to Fig. (9) and indicate with:
pa = feeding pressure
p = operating pressure
Ra = feeding resistance
R = resistance supplied by the pad
Q = oil flow
Fig 9
Given the value of load P and pad dimensions, we may consider with good approximation the operating pressure p=P/Aeff where the denominator term represents the effective area that is the area defined by the average line of the edges, indicated in the figure by means of brushstrokes and of perimeter L.
We consider now what happens in the circuit in case of constant pressure feeding
If we increase the load on the pad, the film height ‘ h ‘ decreases and on the contrary it increases if the load decreases. The pad resistance will then changes depending on the load, as we can also see in the resistance expression:
R = $\bf\frac {12{\large \mu} a}{ h^3 L } = \bf\frac {{\large \Delta} p}{ Q }$ ( v. fig. 9)
$\mu$ = Dynamic viscosity
$\Delta$p = Pressure head astride the pad
Therefore if P increases h decreases and R increases. The total circuit resistance given by R+Ra is bigger and since the feeding pressure ‘pa’ is steady, the delivery through Ra decreases as well, and the pressure ‘p’ under the pad will increase in order to compensate the overload. It is natural to underline that the pressure Pa has to be higher than the pressure that can be generated under the worst operating conditions. From a practical point of view, the feeding resistance can be realized by using a reducing orifice or capillary pipes. It is considered appropriate to have an oil film thicker for about 10 times than the maximum superficial roughness of the bearing surfaces. For what concerns the cavity of the pad, in order not to influence the oil motion and to keep the pressure constant, it must be at least 50-100 times higher than the film.
We consider now the feeding at constant delivery
From a general point of view, each pad shall be fed by a pump without the interposition of any regulation or control device. In this way we will have the pressures diversified from one pad to the other one and the film thickness will remain stiff. The pad acts as a viscous throttling device placed on the pump delivery and thanks to its height it regulate the delivery pressure. A decrease of the film height, due to an increase of the load acting on the pad, will generate an increase of the pressure inside the circuit, and vice versa an increase of the oil film, reducing in this way the R resistance of the pad, will cause a pressure decrease. In practice, to obtain a constant delivery source it is necessary:
• to use single pumps
• to use flow regulators
Flow regulators can be easily found on the market and their installation is obviously simpler than that of pumps. Here below the formula, very simple, that defines the stiffness in case of constant flow:
$K = 3\bf\frac {\large P}{\large h}$
On the contrary the expression relating to the pad feeding at constant pressure is quite more complicated and moreover it depends on the ratio Ra/R (feeding resistance and pad resistance). In case of use of hydrostatic bearings in the hydroelectric field, it is necessary not to forget the high speeds of sliding between runner and pads. This implies the necessity to keep into consideration the increase of delivery to be supplied to the pads because of the removal of oil made by the runner itself. The film thickness, in case of hydrodynamic pads depends on the relative runner/pad speed and on the oil viscosity. In case of hydrostatic feeding, the film thickness doesn’t depend on the viscosity and, theoretically and in extreme case, there is no prohibition to use water as fluid.
The runner/pad relative speed has anyway the tendency to make the pad tilting due to the hydrodynamic effect, and this is the reason why the supporting area is generally partialized Fig. (10) in order to prevent the fluid from outgoing from the pad in a not uniform way. Because of the hydrostatic pad surface design, the resulting slope is much smaller than the one corresponding to the hydrodynamic one.
Fig 10
Here below a table comparing the characteristic values of a traditional bearing installed on one 40 MW turbine as well as the corresponding values, merely theoretical, when using a hydrostatic bearing. Obviously, the power generated by the bearing affects negatively the global efficiency of the machine.
DescriptionHydrodynamicHydrostatic
Rotation speed (Rpm)500500
Average rate pressure on pad (Kg/cm^2)31,2596,5
Oil typeISOVG 46ISOVG 22
Pads average operating temperature (°C )6055
Oil viscosity at operating temperature ( cSt )16,4212,75
Friction coefficient0,002760,0006094
Total oil supply for pads ( lt/min)oil tank287,8
Supply pump pressure ( Kg/cm^2 )63,2
Supply pressure/operating pressure ratio2
Calories developed / absorbed at full capacity ( Cal/s )25,613,37
Calories developed / absorbed at full capacity ( Kw)107,255,9
Oil film min height between pad and rotating part ( mm )0,0440,045
GUIDE BEARING
Again speaking of hydraulic turbines, it is possible to employ the hydrostatic bearings, at least from a theoretical point of view, on condition that appropriate precautions are taken. As it is well known, when using either the guide bearing with pads or those with bushings, the radial difference between the inner diameter of the pad and that of the shaft is certainly higher than a tenth of a illimeter, and even more, depending on the dimensions. It must be understood that this gap is too important for a hydrostatic bearing because it would imply the use of an excessive quantity of oil and a very low stiffness. One solution that can be adopted is to create a load outside the pad by using, for example, the Belleville springs. In this way the pressure of the injected oil will create the quantity of oil film deemed necessary, anyway amounting only to few tens of microns.
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2019-08-18 02:29:54
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https://testbook.com/question-answer/a-hollow-shaft-of-16-mm-outside-diameter-and-12-mm--5f2563d034f3ba0d08b8d0de
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# A hollow shaft of 16 mm outside diameter and 12 mm inside diameter is subjected to a torque of 40 N-m. The shear stress at the outside and inside of the material of the shaft are respectively
This question was previously asked in
ESE Civil 2014 Paper 1: Official Paper
View all UPSC IES Papers >
1. 62.75 N/mm2 and 50.00 N/mm2
2. 72.75 N/mm2 and 54.54 N/mm2
3. 79.75 N/mm2 and 59.54 N/mm2
4. 80.00 N/mm2 and 40.00 N/mm2
Option 2 : 72.75 N/mm2 and 54.54 N/mm2
Free
CT 3: Building Materials
2962
10 Questions 20 Marks 12 Mins
## Detailed Solution
Concept:
Shear stress-induced in the shaft at any location (r) from the centre is given by,
$$\frac{\tau }{r} = \frac{T}{{{I_P}}}$$
Where, r = Radial distance from centre to the location at which shear stress is calculated (m), IP = Polar moment of inertia (m4) and τ = Shear stress N/m2
Polar moment of inertia for hollow circular shaft,
$${I_p} = \frac{\pi }{{32}} \times \left( {D_o^4 - D_i^4} \right)$$
Where, Do = External diameter and Di = Internal diameter of shaft
Calculation:
External diameter (do) = 16 mm
Internal Diameter (di) = 12 mm
Torque Applied (T) = 40 N-m
Polar moment of inertia of hollow circular shaft –
$${I_{P}} = \frac{\pi }{{32}}\left( {{{16}^4} - {{12}^4}} \right) = 4396\;m{m^4}$$
Shear stress at inside of material,
$${\tau _{inside}} = \frac{T}{{{I_P}}} \times {r_i}$$
$$= \frac{{40}}{{4396}} \times \frac{{0.012}}{2}$$
$$= 54.595\; \times {10^{ - 6}}\;N/{m^2} \approx 54.595\;MPa$$
Shear stress at outside of material,
$${\tau _{inside}} = \frac{T}{{{I_P}}} \times {r_o}$$
$$= \frac{{40}}{{4396}} \times \frac{{0.016}}{2}$$
$$= 72.793 \times {10^{ - 6}} \approx 72.793\;MPa$$
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2021-10-19 18:59:13
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https://math.stackexchange.com/questions/linked/1568696
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9k views
237 views
### If $\sum a_n$ is a convergent series with $S = \lim s_n$, where $s_n$ is the nth partial sum, then $\lim_{n \to \infty} \frac{s_1+…+s_n}{n} = S$ [duplicate]
Let $\sum a_n$ be a convergent series, and let $S = \lim s_n$, where $s_n$ is the nth partial sum. I need to prove the following: $\lim_{n \to \infty} \frac{s_1+...+s_n}{n} = S$ How do I go ...
46 views
### Prove that, if $\lim_{j\rightarrow\infty}a_j=l$ then $\lim_{j\rightarrow\infty}m_j=l$.
So I was wondering if anybody could help me tackle this problem that I've been assigned in my real analysis course (book used is Foundations of Analysis by Steven G. Krantz). From my understanding, ...
Be $(a_n)_{n\in N}$ a sequence in $R$ and $a\in R$. Show that $\lim_{n\to \infty}a_n=a ~~~~~\Rightarrow ~~~~~\lim_{n\to \infty} \left(\frac1n \sum_{k=1}^n a_k\right) = a$
We understand that Since $(a_n)$ is convergent hence it's bounded. Let $|a_n|\leq K\ \forall\ n\in \mathbb{N}$. Now for $\epsilon >0$ let $N\in \mathbb{N}$ be such that \$|a_n-A|<\epsilon\ ,\...
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2019-08-24 07:51:46
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https://www.gamedev.net/forums/topic/541664-making-maze-game-in-masm-assembly/
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# Making maze game in MASM (Assembly)
This topic is 3129 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.
## Recommended Posts
I have been working with assembly for some weeks now and I want to actually take on a challenge that actually might interest me in the means to learning it better, so I thought I would try and make a maze game something simple, not sure on size yet I will have to see what my console size is and go from there, I would like some suggestions or maybe a tutorial or example I could get a good idea from and work with. Anything really would help on this, thank you!
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Will you be using real mode assembly (16 bit) and call DOS/BIOS interrupts, or will you go full 32-bit and invoke C functions?
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What part(s) of it don't you think you'll be able to figure out on your own? Try doing a little design work first.
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I am making this entirely ASM since that is what I am working on learning I am trying to stay away from using any C/C++ implementations. I was thinking about a 20x20 character size maze with 1 path through, and I have no idea where to really start I already made a maze on paper it really isn't a great maze but that is ok I just want to try and make something semi interesting to me while I learn this stuff, I would really hate to back away from it, I have been doing C++ for awhile and thought I would in my spare time try and learn some ASM and got a book. Well I am sick of the dumb examples in the book and would like to take a different approach to it. Any ideas on how I should start this, or tutorials out there that I could learn from and get a better understanding.
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Ok I would say this how can I display multiple characters on the screen without making a gotoxy macro then telling what character to write to the screen for every single character space. something that is easy I guess or the best way to do it.I think that is where I should probably start I would then think I would create boundaries after that and create the "character" character say a # is the character, then probably take in input from the user like using the arrows keys. But that is all in the future I just want to be able to display all the walls without typing 1000's of lines of code or even 100's because I dont know the efficient way to do it.
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I did a maze game in ASM in school a few years ago.
http://www.fing.edu.uy/inco/cursos/arqsis2/laboratorio/hist/2004/arqsis2_oblig1_2004.pdf
that's the game we had to do, of course it would be better if you understood spanish for reading it but it has some tips to start with the game.
Good luck.
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Thanks for that ill try and find a program that will translate it, I think google has a document translator not sure, anyways I got my menu made that I liked for now, I might have to change it depending on how the maze goes itself. For my maze it is going to be static so I don't want a to generate like a seed map, this is just for a learning process so I want to avoid anything to complicated that is why I choose a 20x20 map size for now. So for now I guess what I need to know and maybe ill see a good example in the document I haven't looked at yet, is how to write all the walls (Characters) to the screen in an efficient way without having 100's of lines of code to display it, maybe like using a struct or an array, but I am not sure how to go about it.
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ASM is very much like non-object oriented C code (or the opposite way). I once had to write John Conway's Game of Life in ASM. Using the console in Win32 ASM proved mind numbing and I decided to use OpenGL for display.
I used these tutorials -> they are in Russian BUT the code builds step by step and is very similiar to OpenGL in C http://www.pouet.net/topic.php?which=1609.
You could probably steal most of the code and have a conditional jump to draw rectangles in the main program loop.
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thanks for the site ill check it out, so far I have my whole program written except the game itself I created a menu system that takes name, has an option choice that allows you to choose your timer for difficulty and help screen that tells you how to play the game. What I need to do know is,
1.Draw the maze
2.Make the game accept user input from keyboard for the arrow keys
3.Figure out how to create a timer that updates every second
4.Make some sort of collision between the maze walls and user character
So far I know that I can use a macro written for gotoxy to place a solid square
if looks like this,
mGotoxy 0,0 ; puts cursor on x, y positionmov al, 0DBh ;puts a solid square in to the al registercall writechar ;writes the character from al to console
I just don't know how to make that draw across the screen and down below it. Maybe that isn't even a good way or not a possible way to do it, I am just not sure.
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Ok so I know this isn't completely right but for now I am mainly trying to work on movement so I drew a hypothetical maze using a macro I have made for writing to the console. I can't seem to get it to work right and my lack of knowledge is making it hard for me to see the problem here, any help?
Right now the loop ends for somereason even though the if statement isn't true, and because of that my character isn't moving, I know where it is at right now it should leave a trail of _char until I create a cleanup method as well. For now I just wanna get the loop working right and the _char to move around which is a square 0DBh is the hex for it.
Here is my main code and variables:
_x BYTE ? ;x location_y BYTE ? ;y location_char BYTE 0DBh_move DWORD ?
;*****************GAME LOOP*********************DRAW: mWrite "#################X##" mWrite "################ ##" mWrite "################ ###" mWrite "########### ###" mWrite "########### ########" mWrite "########### ### ##" mWrite "########### ### ####" mWrite "########### ####" mWrite "############ #######" mWrite "#### #######" mWrite "############ #######" mWrite "############ #######" mWrite "############ #######" mWrite "############ #######" mWrite "####### #######" mWrite "####### ############" mWrite " ############" mWrite "####### ############" mWrite "####### ####" mWrite "####################" mov _x, 0 mov _y,16 JMP GAMEGAME: call readstring mov _move, eax call writeint mGotoxy _x, _y mov al, _char call writechar .if _move == 119 ; UP dec _y .endif .if _move == 115 ; DOWN inc _y .endif .if _move == 97 ; LEFT dec _x .endif .if _move == 100 ; RIGHT inc _x .endif .if _x == 1 || _y == 1 JMP OVER .endif JMP GAME ;*****************END GAME**********************OVER: exit
EDIT: BTW I cannot use interrupts, so please don't suggest it. Vista doesn't allow that kind of access to the bios.
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2018-02-18 11:13:01
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https://www.physicsforums.com/threads/estimate-the-total-harmonic-distortion-present.882954/
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# Estimate the total harmonic distortion present
1. Aug 24, 2016
### Hndstudent
• Thread moved from the technical forums, so no Homework Help Template is shown.
The supply current was sampled 1024 times over a very short time interval. The data so obtained is given in column B of the accompanying Excel worksheet1. This worksheet has been set up to give a graph showing the spectral components of the data.
Question 3
1. i) Obtain the Fourier Transform for the data using the Fourier Analysis tool of Excel. The transformed data should commence in cell D2.
2. ii) Identify the principal frequencies in the current waveform.
3. iii) Estimate the total harmonic distortion [THD] present in the current waveform using the formula:
n max
THD(I)=1/I1 SQRT Σ (In)^2 x 100%
n=2
where I1 is the r.m.s. value of the fundamental current, In the r.m.s value of the nth harmonic and n(max) is the number of the highest measurable or significant harmonic.
[Note the vertical axis of the spectrum graph is scaled in (current)2.]
iv) Attempt to synthesise the shape of the original waveform from its principal harmonics [e.g. sketch the waveforms of the harmonics on the same time axis and add them together].
I am struggling with 3iii.
for I1 rms I have SQRT 15.8 + 8.77 + 6.25 = 5.55A
n max I have 15.8 as its the highest magnitude.
But I'm not sure what to do to find In?
any help would be appreciated.
2. Aug 25, 2016
### Staff: Mentor
In, or $I_n$ to make it clear, should be the rms magnitudes of the individual harmonic components.
Perhaps you could attach the Excel spreadsheet, or even a textfile of the datapoints you were given so that others might duplicate the problem?
3. Aug 25, 2016
### Hndstudent
Hi Gneill,
please find attached original worksheet. and a screen shot of the completed waveform.
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4. Aug 26, 2016
### Hndstudent
Hi Gneill, I am still struggling to work out $I_n$. And am I close with my I1 rms and n max answers?
Thanks
5. Aug 26, 2016
### Staff: Mentor
$I_1$ will be the magnitude of the fundamental alone. That is, the single peak that corresponds to the fundamental frequency associated with the current. In general this is the peak with the lowest frequency. The other peaks should be located at some multiples of that frequency. In your data the fundamental corresponds to the 15.878 at a frequency of about 53 Hz.
$n_{max}$ is the number of the frequency component peak with the highest frequency that you intend to deem as "significant". The peaks are numbered from 1 to $n_{max}$, with "1" being associated with the fundamental. If you plot your FFT values, how many significant peaks can you see over the whole domain? It looks like you've already picked them out and there are three of them. So $n_{max}$ = 3.
As an aside, I have noticed that different implementations of the FFT algorithm tend to apply different normalizing factors to the returned values. You might find, for example, that in order to recover the actual component contributions to your "signal" that you have to multiply the returned values by some constant, typically 2. The "DC" contribution (if any) is usually exempt from this normalization (I don't know why this is).
As an exercise you might try concocting your own "signal" with known components at particular frequencies and see how your Excel FFT handles it. Maybe something like:
$f(t) = 10 sin(ω_o t) + 3 sin(5 ω_o t) + 2 sin(8 ω_o t)$
where you choose the fundamental frequency $ω_o$ in radians/sec as you wish. Also pick a suitable sampling frequency to "sample" the signal generate the raw datapoints. See if you can "recover" the harmonic frequencies and their magnitudes from the FFT data.
6. Mar 23, 2017
### David J
Hi, is this thread still active as I have a question relating to this topic. Please advise or should i start a new thread ?? Thanks
7. Mar 23, 2017
### Staff: Mentor
Probably best to start a new thread. It's unlikely that the OP is still interested in pursuing this thread at this time.
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2017-11-22 02:31:54
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https://yourgolfermagazine.co.uk/blog/cuq2oq.php?97feb0=voltage-divider-rule-in-parallel-circuit
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Similarly, Current Divider Rule (CDR) shows how current distributes in a parallel circuit. Single and Double Subscript Notation, 4.9. As you know, there are two types of combinations in a circuit, they are series and parallel connections. Voltage Divider Rule (VDR) shows how the voltage distributes among different resistor in a series circuit. The divide-down ratio is determined by two resistors. Internal Resistance of Voltage Sources, 6.6. Brief about Building Automation Systems, 10 LED Lighting electronic project ideas for fun, The power supply voltage divides fractionally in a series electrical circuit, The power supply current divides in parallel electrical circuit partly, VDR is used to find the partial voltage across the individual resistor in a series circuit, CDR is used to find the partial current in the individual resistor in a parallel circuit. Michael has got his undergraduate degree in 2016 from a reputable university securing high grads. The proportionality between different branch currents is strictly a function of resistance. Can I close my money in HSA and transfer it to another bank account and use it for non-medical purpose? Ammeter, Voltmeter and Ohmmeter Design, $$\bbox[5px,border:1px solid red] {\color{blue}{{I_1 \over I_2}= Color coding and Standard Resistor Values, 4.6. Is there a model of ZFC that can define a "longer" model of ZFC to which it is isomorphic? Calculating R1 and R2 . The reason this difference is the difference in resistance. For the circuit given, suppose we are interested in current I3 in R3 and we know the total current of the circuit from the above calculation. the current through any branch of a parallel resistive network is In a series circuit ,the voltage across the circuit is the sum of the voltages across each component. R1 = R4 = 1k ohm equal to the total resistance of the parallel network divided by the site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. The Current Divider Rule Calculator, calculates the current in Rx only based on current divider formula. R5 = 10 ohm By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. But current divides such that the summation of individual resistor current is always equal to source current. Podcast 282: Stack Overflow’s CEO reflects on his first year, Parallel and Series Circuits Question (Electron Flow), When use Voltage divider in circuit analysis. It's output voltage is a fixed fraction of its input voltage. Calculating sum resistance of multiple series and parallel resistors. While the current for series circuit remains same throughout the divider circuit as discussed earlier. Note that V3 is same as we calculated in the previous section using Ohm’s law. This means that the current going through the load is ten times the … The formula for current divider is: I1 is the branch current where R1 is connected and we are interested in, IT is the total current provided by the source and RT is the total resistance of the parallel resistor circuit. of the voltage divider: • R TH: 2 100k in parallel, 100k/2 = 50k • V • Measure voltage across R in (V out)= 8V, thus 2V drop across R TH • The relative size of the two resistances are in proportion of these two voltage drops, so R in must be 4 (8/2) R TH, so R in = 200k TH =20 100k 2×100k 1V How I should get the voltage through voltage divider in figure 1 and how the voltage changes for R1, R2, R3 and R4 in figure 2 (when the wire for current isn't there anymore). The calculation will be:V_{2}=\frac{140 v\times 40\Omega }{70 \Omega }$$V_{2}=80 v$. (R1+R2)//(R3+R4) , therefore voltage across R1+ R2 are same as R3+R4, find this voltage V, then use voltage divider rule again . What is the difference between число and количество? If you found voltage across R1+R2 , if R1//R2 then V is same for them. By looking closely to these number, you will observe that the voltage drop is different from each other and the summation of all of them is equal to the voltage applied to that circuit (source). Determine the size of the resistor used in the voltage divider circuit using the 10% rule of thumb. That's one way to solve this. Voltage Divider and Current Divider are the most common rules applied in practical electronics. HOME TABLE OF CONTENTS Are Java programs just instances of the JRE? Electrical Engineering Stack Exchange is a question and answer site for electronics and electrical engineering professionals, students, and enthusiasts. ratio equal to the inverse of their resistance values. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. The VDR formula for V3 will be:$V_{3}=\frac{V_{T} R_{3}}{R_{Eq}}$, By putting the corresponding values, we get:$V_{3}=\frac{140 v\times 10\Omega }{70 \Omega }V_{3}=20 v\$. The Overflow #45: What we call CI/CD is actually only CI. Voltage Division in a Series Circuit, 4.8.1. CDR is the counterpart in a parallel electric circuit to VDR in series electric circuits. sum of the two resistances. In a parallel resistors, the voltage across each resistor is same as the source voltage. resistance of the resistor of interest and multiplied by the total current Why does the US death rate not "match" life expectancy. Is creation of new states via partitioning really possible in the US? Thanks for answering so fast. For left figure : R1//R3 => same voltage across R2//R4 => same voltage across, Now use voltage divider rule (apply for resistors in series), You have R5 in series with (R1//R3) and with (R2//R4), For right figure you have R5 in series with [(R1+R2)//(R3+R4)], use voltage divider rule to find V across [(R1+R2)//(R3+R4)]. Find out the total current and the current flowing through each resistor in the parallel circuit using the current division rule. Encyclopedia of Electrical Engineering. The two resistor voltage divider is used often to supply a voltage different from that of an available battery or power supply. Introduction to Microcontroller and its Benefits, What is Smart Building?
Brunch Dublin, I Love Sushi Seattle, Marshall Dsl100h Tubes, What Vegetables Like Eggshells, How To Fight Lunastra, Peripheral Artery Disease Causes, Blackstar Ht Club 40, Throng Antonym,
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2021-08-06 02:25:26
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http://mathhelpforum.com/differential-geometry/111861-homeomorphism.html
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# Math Help - Homeomorphism
1. ## Homeomorphism
Show that (a,b) is homeomorphic to R..
Thank you in advance
2. Originally Posted by math.dj
Show that (a,b) is homeomorphic to R..
Thank you in advance
You've left out a lot! Are we to assume that (a, b) is an interval in R with the "usual topology"? If so, you basically need to find a one-to-one function from (a,b) onto R. You should be able to find a rational function that maps a to $-\infty$ and b to $\infty$.
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2014-04-18 22:59:14
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https://math.stackexchange.com/questions/3476746/why-is-this-piece-wise-function-differentiable-at-the-point-x-0
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# Why is this piece-wise function differentiable at the point $x=0$?
I am given the following function:
$$f : \mathbb{R} \rightarrow \mathbb{R} \hspace{2cm}$$
$$f(x) = \left\{ \begin{array}{ll} x^2 \sin(\frac{1}{x}) & \quad x \neq 0 \\ 0 & \quad x = 0 \end{array} \right.$$
And I do not understand why this function is differentiable in $$x = 0$$, as my textbook claims. Firstly, I know that the function must be continuous in $$x=0$$ for us to even discuss the possibility of it being differentiable. For the function to be continuous at $$x=0$$, the limits of the function from both sides of $$x=0$$ must equal the value of the function itself at $$x=0$$. We have:
$$f(0)=0$$
$$\lim\limits_{x \to +0}f(x) = \lim\limits_{x \to +0} x^2 \sin\bigg (\frac{1}{x} \bigg ) = 0$$
$$\lim\limits_{x \to -0}f(x) = \lim\limits_{x \to -0} x^2 \sin\bigg (\frac{1}{x} \bigg ) = 0$$
So we can see that the function is indeed continuous at $$x=0$$. Now it is natural to discuss differentiability. For the function to be differentiable at $$x=0$$, the limit of the derivatives from both sides of $$x=0$$ must be equal to the derivative itself at $$x=0$$. If I find the derivative of $$f(x)$$ I get:
$$f'(x) = \left\{ \begin{array}{ll} 2x \sin(\frac{1}{x}) - \cos(\frac{1}{x}) & \quad x \neq 0 \\ 0 & \quad x = 0 \end{array} \right.$$
So
$$f'(0) = 0$$
$$\lim\limits_{x \to +0}f'(x) = \lim\limits_{x \to +0} 2x \sin \bigg( \frac{1}{x} \bigg) - \cos\bigg ( \frac{1}{x} \bigg )$$
$$\lim\limits_{x \to -0}f'(x) = \lim\limits_{x \to -0} 2x \sin \bigg( \frac{1}{x} \bigg) - \cos\bigg ( \frac{1}{x} \bigg )$$
And the last $$2$$ limits do not exist because of the $$\cos(\frac{1}{x})$$ term.
So I am really confused as to why this function is differentiable at $$x=0$$. What mistakes did I make in my reasoning? Why is the derivative of the function at $$x=0$$ well defined?
• While it is true that differentiability at $0$ implies continuity at $0$, that does not oblige you to first verify continuity at $0$ before you go on to consider differentiability at $0$. And, in fact, once you have demonstrated differentiability at $0$, you may conclude continuity at $0$ without any further work. That's how implications work. – Lee Mosher Dec 15 '19 at 1:32
## 2 Answers
"Differentiable at $$0$$" does not mean that the derivative is continuous, but rather that the derivative exists at $$0$$. You can see it by taking the limits of the Newton quotients at $$0$$: $$\frac{h^2\sin\frac1h}{h}=h\sin\tfrac1h\xrightarrow[h\to0]{}0,$$ so the derivative exists at $$0$$ and is $$0$$.
The "squeezing" by $$x^2$$ is what makes the function differentiable at $$0$$:
Diffentiability of $$f$$ at $$0$$ doesn't guarantee the continuity of $$f'$$ at $$0$$, and this is a classic example.
Indeed, the reasoning of the differentiability of $$f$$ at $$0$$ is the following.
For $$h\ne 0$$, $$|f(h)-f(0)|=|f(h)|=|h^{2}\sin(1/h)|$$, so $$|f(h)/h|\leq|h||\sin(1/h)|\leq|h|\rightarrow 0$$, therefore $$f'(0)=0$$.
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2021-08-01 21:42:13
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https://quant.stackexchange.com/questions/43873/alternative-method-for-determining-option-implied-pdf
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# Alternative Method for Determining Option-Implied pdf
As I am refining a pricing model to incorporate skew, and not just ATM volatilities, I need to create random realizations of the underlying consistent with the skew-implied pdf. When searching, one ends up with the Breeden-Litzenberger formula, which states that:
$$\frac{\partial^{2}C}{\partial K^{2}}=e^{-rT}g(S_{T})$$
As I am slightly wary of using numerical second derivatives in my code, I looked for an alternative way of obtaining this. I came up with the idea of using correctly skew-adjusted cash-or-nothing binary put prices in order to derive the CDF of the underlying:
$$P_{dig}=P_{dig,noskew}+\nu_{vanilla}*\frac{\partial \sigma}{\partial K}$$
As $$CDF=e^{rT}P_{dig}$$, I run [0,1]-normally distributed random numbers through the inverse function $$CDF^{-1}$$ to get realizations of the underlying that are distributed consistent with the skew.
Can I please have your views on this / please comment if I am missing something here
• If all you want is the CDF and not the PDF, then you could also use $\partial C / \partial K$ instead of the second order derivative. You might also find my answer to quant.stackexchange.com/questions/30749 helpful. – LocalVolatility Feb 3 '19 at 22:30
• So basically that would be $1+\partial C/\partial K$ – ZRH Feb 4 '19 at 7:58
• Yes - but don't forget to multiply the derivative by the discount factor. – LocalVolatility Feb 4 '19 at 9:19
• Coded this up now as proposed by you, works like a charm. For anyone who wants to try - you need to compute the CDF over a quite wide range of values in order not to truncate the pdf that is implicit in the distribution of underlying values. – ZRH Feb 4 '19 at 16:16
• @zrh if you have some analytic form for the smile, then you can compute the second derivative numerically to arbitrary precision - what is your worry? – will Feb 5 '19 at 21:47
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2020-11-27 03:45:19
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https://physics.stackexchange.com/questions/149018/does-the-lagrangian-contain-all-the-information-about-the-representations-of-the
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# Does the lagrangian contain all the information about the representations of the fields in QFT?
Given the Lagrangian density of a theory, are the representations on which the various fields transform uniquely determined?
For example, given the Lagrangian for a real scalar field $$\mathscr{L} = \frac{1}{2} \partial_\mu \varphi \partial^\mu \varphi - \frac{1}{2} m^2 \varphi^2 \tag{1}$$ with $(+,-,-,-)$ Minkowski sign convention, is $\varphi$ somehow constrained to be a scalar, by the sole fact that it appears in this particular form in the Lagrangian?
As another example: consider the Lagrangian $$\mathscr{L}_{1} = -\frac{1}{2} \partial_\nu A_\mu \partial^\nu A^\mu + \frac{1}{2} m^2 A_\mu A^\mu,\tag{2}$$ which can also be cast in the form $$\mathscr{L}_{1} = \left( \frac{1}{2} \partial_\mu A^i \partial^\mu A^i - \frac{1}{2} m^2 A^i A^i \right) - \left( \frac{1}{2} \partial_\mu A^0 \partial^\mu A^0 - \frac{1}{2} m^2 A^0 A^0 \right). \tag{3}$$ I've heard$^{[1]}$ that this is the Lagrangian for four massive scalar fields and not that for a massive spin-1 field. Why is that? I understand that it produces a Klein-Gordon equation for each component of the field: $$( \square + m^2 ) A^\mu = 0, \tag{4}$$ but why does this prevent me from considering $A^\mu$ a spin-1 massive field?
[1]: From Matthew D. Schwartz's Quantum Field Theory and the Standard Model, p.114:
A natural guess for the Lagrangian for a massive spin-1 field is $$\mathcal{L} = - \frac{1}{2} \partial_\nu A_\mu \partial_\nu A_\mu + \frac{1}{2} m^2 A_\mu^2,$$ where $A_\mu^2 = A_\mu A^\mu$. Then the equations of motion are $$( \square + m^2) A_\mu = 0,$$ which has four propagating modes. In fact, this Lagrangian is not the Lagrangian for a amassive spin-1 field, but the Lagrangian for four massive scalar fields, $A_0, A_1, A_2$ and $A_3$. That is, we have reduced $4 = 1 \oplus 1 \oplus 1 \oplus 1$, which is not what we wanted.
• Why would writing out the Lagrangian for each component of a vector field prevent you from viewing the vector field as a vector field? I think whatever you heard about not being able to do so is wrong. – zzz Nov 26 '14 at 17:21
• Also re the title, at least within the scope of what you're asking, the Lagrangian specifies the representation by the virtue that it is written in terms of a field in some specific rep, e.g. a scalar field Lagrangian specifies dynamics of a scalar field not a vector one. But of course that says nothing about not being able to view components of a vector field as scalar fields – zzz Nov 26 '14 at 17:28
• If each component of A satisfies the Klein-Gordon equation, that doesn't necessarily mean that the components of A transform like a vector under Lorentz transformations. – jabirali Nov 26 '14 at 18:05
• Comment to the question (v5): As M. Schwartz mentions on top of p. 115, the energy density for the Lagrangian (2) is not bounded from below because the kinetic term of the $A_0$ field has the wrong sign, and hence the theory is not physical in the first place. Therefore the discussion of possible representations and interpretations of (2) seems somewhat academic. On the other hand, if $A_0$ did not have the wrong sign, then $A_{\mu}$ could not be viewed as a 4-covector, but could only be interpreted as 4 scalars. – Qmechanic Nov 26 '14 at 20:21
• @glance, "why can't I say (or can I?) that Aμ is a spin-1 field for that choice of the Lagrangian?" Yes you definitely can, then the reason to reject (2) becomes the energy bounded below condition, instead of the one Schwartz gave. Qmechanic's comment is right on. – Jia Yiyang Nov 27 '14 at 13:20
What Qmechanic said in comments is pretty solid, "Lagrangian (2) is not bounded from below because the kinetic term of $A_0$ field has the wrong sign, and hence the theory is not physical in the first place", but I think your Question needs a change of emphasis. Your Lagrangian allows us to construct four equations of motion for four non-interacting fields. That the kinetic energy is not bounded below doesn't matter for a classical field theory if there are no interactions. We can say that $A_\mu$ transforms as a Minkowski 4-vector. But we can't do any significant classical physics with it because any Lorentz invariant interacting system would be unstable, and there have to be interactions of some sort (for us to be able to discuss measurement of the field by using its effects on other fields, for example, rather than just to talk about it as a theoretical object).
In the QFT context, however, we have to construct a positive semi-definite inner product (over the test function space, essentially the creation/annihilation operator commutator has to be positive-semidefinite in the 4-momentum coordinate space) for us to be able to construct a Fock space, which we cannot do for Lagrangian (2) even if there are no interactions. That is, QFT imposes an additional requirement even for a free quantum field, which forces us to the Proca equation or to the Maxwell equation for spin 1, because as well as a dynamics we also need a probability interpretation for observables (which is what the Hilbert space structure gives us).
EDIT: For local symmetries, if we use an n-dimensional vector space at each point, then the structure is defined by the ways in which we use the metric or other multilinear forms to construct a positive semi-definite inner product over the vector space at a point. If we use the metric to construct terms that can be written using the Lorentz metric, such as $g^{\mu\nu}A_\mu A_\nu$, then the field at a point can be taken to be a vector representation of the Lorentz group; if we use the metric to construct multinomials in $A_{\mu\nu}$, then we have a field that has a tensor structure, etc. If instead we introduce a different constant bilinear tensor $h^{\alpha\beta}$ and use the form $h^{\alpha\beta}A_\alpha A_\beta$, then the field is a vector representation of whatever symmetries $h$ has; if we use some higher degree multinomial in components $A_a$, then the field is a representation space of whatever the symmetries are of that higher degree multinomial.
Remembering, however, that we have to have a positive semi-definite energy for Physics, which requires work when we start from an indefinite metric.
There is also a global aspect of the Hilbert space, in the quantum case, that is not determined by the Lagrangian, at least insofar as one also has to specify the vacuum state, a thermal state, or some other state as a lowest energy state.
• thanks for the answer. However, you focused on a particular example I provided. Can you also say something about the main question (and the other example I gave)? – glS Nov 27 '14 at 20:24
• @glance Right, and apologies. Read the Question carefully. I take it that there are two aspects to the question: the local and the global. See my Edit, to appear momentarily, I hope. – Peter Morgan Nov 28 '14 at 23:13
Field $\psi_{a_{1}...a_{n}\dot{b}_{1}...\dot{b}_{m}}$ with a given spin and mass (i.e. field which transforms under irrep of the Poincare group) must satisfy some determined conditions called irreducibility conditions: $$\tag 1 \hat{W}^{2}\psi_{a_{1}...a_{n}\dot{b}_{1}...\dot{b}_{m}} = -m^{2}\frac{n + m}{2}\left(\frac{n + m}{2} + 1\right)\psi_{a_{1}...a_{n}\dot{b}_{1}...\dot{b}_{m}},$$ $$\tag 2 \hat{P}^{2}\psi_{a_{1}...a_{n}\dot{b}_{1}...\dot{b}_{m}} = m^{2}\psi_{a_{1}...a_{n}\dot{b}_{1}...\dot{b}_{m}}.$$ Here $\hat{W}$ is Pauli-Lubanski operator and $\hat{P}$ is translation operator. Representation with equal quantity $\frac{n + m}{2}$ are equivalent.
As for details, look here for the information about indices and Lorentz representations and here for fields as representations of Poincare group.
If you construct lagrangian which leads to $(1), (2)$, you will uniquely determine transformation properties of field with a given mass and spin.
• could you elaborate a bit? What kind of field is $\psi$? Why did you used different kind of labels (some dotted and some not) for its indices? Can you also give some reference for your statements (or show their validity)? Finally, you didn't directly addressed the point of the question: how does the Lagrangian lead to the representation of the fields you are mentioning? – glS Nov 27 '14 at 21:51
• @glance : as for the references, I've added some into the answer. As for the last question, the answer is following. Lagrangian is secondary quantity in QFT formalism, because equations of motion are obtained by using requirement of irreducibility of representation which is realizes by field. Equations of motion determine the representation uniquely. Lagrangians are constructed by the way that they must lead to the equations of motion, which determine the representation uniquely. – Andrew McAddams Nov 28 '14 at 8:31
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2019-08-21 22:34:53
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http://askubuntu.com/questions/328059/black-screen-after-13-04-how-do-i-get-the-display-back-through-the-command-line?answertab=active
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# Black screen after 13.04. How do I get the display back through the command line
This is a common problem with Ubuntu. If I open the command line, something pops up really fast that says the BIOS disabled something. It let me login on the command line. I tried a bunch of commands that I thought would have opened or updated my bios. Nothing worked. Is there a way I can get my display working again through the command line?
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It said kmv: disabled by bios – John Aug 3 '13 at 1:22
You mean KVM :disabled by bios right?? – Stormvirux Aug 3 '13 at 2:14
The message about KVM disabled may have nothing to do with your problem. Please re-describe your problem. Are getting a black-screen (with just cursor top right), when you want to see the GUI (graphic interface)? – david6 Aug 3 '13 at 2:30
Have you considered looking into your BIOS settings? There is usually an "Intel VT" or "Virtualization" option that isn't enabled by default. This is being modified. If you are unable to modify it in the Bios Try the following
First of all, you might want to see if KVM is loaded in the kernel. To do this, run something like this:
lsmod | grep kvm
If you do get any answer from that, you may want to stop kvm from running. To do that, this will help:
modprobe -r kvm
Remember to follow for any other kvm module. In my case, there is also a kvm_intel module, so I also have to run modprobe -r kvm_inteland modprobe -r kvm_amd.
To stop kvm from being enabled at boot time you have to find the file located at /etc/modules and modify it so that KVM doesn't launch at boot time. After you modify your /etc/modules file restart your computer and see if the problem persists.
I would recommend checking out the following:
If you do not use VMware you can run the following(It breaks VMWare)
sudo apt-get purge kvm
If you want to still continue using quemu with KVM look for a more recent BIOS on the vendor's web site.
Note:
• On some hardware (e-g HP nx6320), you need to power-off/power-on the machine after enabling virtualization in the BIOS.
• Enabling some BIOS features may break VT support on some hardware (e-g Enabling Intel AMT on a Thinkpad T500 will prevent kvm-intel from loading with "disabled by bios")
• On some Dell hardware, you also need to disable "Trusted Execution", otherwise VT will not be enabled.
-
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2015-01-27 12:59:39
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http://codingadventures.org/2012/04/20/the-missing-file-warning-on-xcode-4/
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# The missing file warning on Xcode 4
If you are in a situation where the files that are being shown have been deleted from the project navigator view but it still seems to be showing up as somehow linked to the project, even though the file no longer appears in the Project navigator, you must remove also from the svn in order to resolve the warnings.
Just open the Terminal utility, browse to your folder where your .h/.m files located and type the following command:
svn delete nameOfMissingFile
Update: Mountain Lion users check this!
1. RB says:
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2021-04-18 05:38:40
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https://paradigms.oregonstate.edu/problem/757/
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## Magnetic susceptibility
• Paramagnet Magnetic susceptibility
• assignment Paramagnetism
assignment Homework
##### Paramagnetism
Energy Temperature Paramagnetism Thermal and Statistical Physics 2020 Find the equilibrium value at temperature $T$ of the fractional magnetization $$\frac{\mu_{tot}}{Nm} \equiv \frac{2\langle s\rangle}{N}$$ of a system of $N$ spins each of magnetic moment $m$ in a magnetic field $B$. The spin excess is $2s$. The energy of this system is given by \begin{align} U &= -\mu_{tot}B \end{align} where $\mu_{tot}$ is the total magnetization. Take the entropy as the logarithm of the multiplicity $g(N,s)$ as given in (1.35 in the text): $$S(s) \approx k_B\log g(N,0) - k_B\frac{2s^2}{N}$$ for $|s|\ll N$, where $s$ is the spin excess, which is related to the magnetization by $\mu_{tot} = 2sm$. Hint: Show that in this approximation $$S(U) = S_0 - k_B\frac{U^2}{2m^2B^2N},$$ with $S_0=k_B\log g(N,0)$. Further, show that $\frac1{kT} = -\frac{U}{m^2B^2N}$, where $U$ denotes $\langle U\rangle$, the thermal average energy.
• assignment Extensive Internal Energy
assignment Homework
##### Extensive Internal Energy
Energy and Entropy 2021 (2 years)
Consider a system which has an internal energy $U$ defined by: \begin{align} U &= \gamma V^\alpha S^\beta \end{align} where $\alpha$, $\beta$ and $\gamma$ are constants. The internal energy is an extensive quantity. What constraint does this place on the values $\alpha$ and $\beta$ may have?
• assignment Exponential and Logarithm Identities
assignment Homework
##### Exponential and Logarithm Identities
Static Fields 2022 (2 years)
Make sure that you have memorized the following identities and can use them in simple algebra problems: \begin{align} e^{u+v}&=e^u \, e^v\\ \ln{uv}&=\ln{u}+\ln{v}\\ u^v&=e^{v\ln{u}} \end{align}
• assignment Nucleus in a Magnetic Field
assignment Homework
##### Nucleus in a Magnetic Field
Energy and Entropy 2021 (2 years)
Nuclei of a particular isotope species contained in a crystal have spin $I=1$, and thus, $m = \{+1,0,-1\}$. The interaction between the nuclear quadrupole moment and the gradient of the crystalline electric field produces a situation where the nucleus has the same energy, $E=\varepsilon$, in the state $m=+1$ and the state $m=-1$, compared with an energy $E=0$ in the state $m=0$, i.e. each nucleus can be in one of 3 states, two of which have energy $E=\varepsilon$ and one has energy $E=0$.
1. Find the Helmholtz free energy $F = U-TS$ for a crystal containing $N$ nuclei which do not interact with each other.
2. Find an expression for the entropy as a function of temperature for this system. (Hint: use results of part a.)
3. Indicate what your results predict for the entropy at the extremes of very high temperature and very low temperature.
• assignment Power Series Coefficients 2
assignment Homework
##### Power Series Coefficients 2
Static Fields 2022 (6 years) Use the formula for a Taylor series: $f(z)=\sum_{n=0}^{\infty} \frac{1}{n!} \frac{d^n f(a)}{dz^n} (z-a)^n$ to find the first three non-zero terms of a series expansion for $f(z)=e^{-kz}$ around $z=3$.
• face Thermal radiation and Planck distribution
face Lecture
120 min.
##### Thermal radiation and Planck distribution
Thermal and Statistical Physics 2020
These notes from the fourth week of Thermal and Statistical Physics cover blackbody radiation and the Planck distribution. They include a number of small group activities.
• assignment Series Notation 2
assignment Homework
##### Series Notation 2
Power Series Sequence (E&M)
Static Fields 2022 (6 years)
Write (a good guess for) the following series using sigma $\left(\sum\right)$ notation. (If you only know a few terms of a series, you don't know for sure how the series continues.)
1. $1 - 2\,\theta^2 + 4\,\theta^4 - 8\,\theta^6 +\,\dots$
2. $\frac14 - \frac19 + \frac{1}{16} - \frac{1}{25}+\,\dots$
• assignment_ind Normalization of the Gaussian for Wavefunctions
assignment_ind Small White Board Question
5 min.
##### Normalization of the Gaussian for Wavefunctions
Periodic Systems 2022
Fourier Transforms and Wave Packets
Students find a wavefunction that corresponds to a Gaussian probability density.
• group Fourier Transform of a Gaussian
group Small Group Activity
10 min.
##### Fourier Transform of a Gaussian
Periodic Systems 2022
Fourier Transforms and Wave Packets
• assignment Zapping With d 1
assignment Homework
##### Zapping With d 1
Energy and Entropy 2021 (2 years)
Find the differential of each of the following expressions; zap each of the following with $d$:
1. $f=3x-5z^2+2xy$
2. $g=\frac{c^{1/2}b}{a^2}$
3. $h=\sin^2(\omega t)$
4. $j=a^x$
5. $k=5 \tan\left(\ln{\left(\frac{V_1}{V_2}\right)}\right)$
• Thermal and Statistical Physics 2020
Consider a paramagnet, which is a material with $n$ spins per unit volume each of which may each be either “up” or “down”. The spins have energy $\pm mB$ where $m$ is the magnetic dipole moment of a single spin, and there is no interaction between spins. The magnetization $M$ is defined as the total magnetic moment divided by the total volume. Hint: each individual spin may be treated as a two-state system, which you have already worked with above.
1. Find the Helmholtz free energy of a paramagnetic system (assume $N$ total spins) and show that $\frac{F}{NkT}$ is a function of only the ratio $x\equiv \frac{mB}{kT}$.
2. Use the canonical ensemble (i.e. partition function and probabilities) to find an exact expression for the total magentization $M$ (which is the total dipole moment per unit volume) and the susceptibility \begin{align} \chi\equiv\left(\frac{\partial M}{\partial B}\right)_T \end{align} as a function of temperature and magnetic field for the model system of magnetic moments in a magnetic field. The result for the magnetization is \begin{align} M=nm\tanh\left(\frac{mB}{kT}\right) \end{align} where $n$ is the number of spins per unit volume. The figure shows what this magnetization looks like.
3. Show that the susceptibility is $\chi=\frac{nm^2}{kT}$ in the limit $mB\ll kT$.
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2022-11-30 21:11:06
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http://learning.maxtech4u.com/2018/07/page/3/
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What is High Performance Computing (HPC)?
Technology & Science / July 4, 2018
High performance computing (HPC) has become increasingly more popular and important in the world of engineering. But defining what HPC is and figuring out how it can be deployed to aid designers can be tricky. Nowadays, high performance computing is more and more important for the economic and technological development. The high performance computing also becomes an indicator to measure the power of a country. Therefore, it is important and meaningful to improve the performance and universality of high performance computing. High-performance computing (HPC) is the use of super computers and parallel processing techniques for solving complex computational problems. Basic Description of High Performance Computing (HPC) High Performance Computing (HPC) most generally refers to the practice of aggregating computing power in a way that delivers much higher performance than one could get out of a typical desktop computer or workstation in order to solve large problems in science, engineering, or business. High Performance Computing (HPC) allows scientists and engineers to solve complex, compute-intensive problems. HPC applications often require high network performance, fast storage, and large amounts of memory, very high compute capabilities, or all of these. High performance computing runs a broad range of systems, from our desktop computers through large…
What is World Wide Web (WWW)?
Technology & Science / July 3, 2018
The WWW is such an immensely popular Internet facility that for many users, it has become synonymous with the Internet. The World Wide Web was established with the objective of accessing the data from anywhere at any time in form of interlinked hypertext language. The internet is a huge network of computers all connected together. The World Wide Web (‘www’ or ‘web’ for short) is a collection of WebPages found on this network of computers. Your web browser uses the internet to access the web. Overview of WWW World-Wide Web (also called WWW or W3) is a hypertext-based information system. Any word in a hypertext document can be specified as a pointer to a different hypertext document where more information pertaining to that word can be found. The reader can open the second document by selecting the word (using different methods depending on the interface; in a mouse based system, a user would probably place the mouse over the word and click the mouse button); only the part of the linked document which contains relevant information will be displayed. The World Wide Web, or just “the Web,” as ordinary people call it, is a subset of the Internet. The Web…
Waterfalls near Indore to visit in Monsoon Season
/ July 2, 2018
Dive deep into the details of destinations, culture and everything else that will make your experience in Madhya Pradesh, a memorable one. Indore. Indore is often known to be a historical city living with its cultural heritage. But equally enchanting are the natural wonders that are bestowed upon it. It is located on Malwa Plateau and this location blesses it with many untamed Waterfalls near indore. Most of these waterfalls are formed mainly during the monsoons but they remain to be the sought after picnic destinations for the adventure loving Indorites and tourists. The beautiful valleys and lush green locales surrounding these cascading waterfalls makes them exotic and picture perfect. They are identical to the city of Indore and are always listed on the must see places for the travelers who come to explore this wonderful and diverse city. Lying on the verge of Malwa Plateau, Indore has many untamed waterfall near indore. A catalog of the attractive waterfalls from the state perhaps doesn’t exist but after much data collection the following list of waterfall near indore have to visit in monsoon season. 1. Patalpani Waterfall Patalpani waterfalls are located near Mhow, very close to Indore. The falls are huge…
Insert math as
$${}$$
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2018-08-18 14:32:19
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https://anarchy.website/
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anarchy.website Toggle Dark Mode
# Anarchy.Website
Welcome to Anarchy.Website, a site built as a ‘blog’ and archive by Una Ada for hosting their written content. While the name of the site implies a focus on anarchism, no such restriction exists, with much of the content here being comedic fiction or notes from classes.
## Recent Posts
### Anarchy.Websiteに振り仮名と地域化
Sleep deprived ranting about ruby text, internationalization, localization, CSS3 support, monospace fonts, and some other stuff. Does this count as documentation? Am I doing a documentation here?
December 11, 2019, tags: #blog #localization #programming #web-dev
### Goth Angel Sinners Pt. V: Post-Scarcity Notes
A couple of sleep deprived losers try to explain the philosophical and ecological necessity and implications of post-scarcity as a quote unquote economic mode.
June 25, 2019, tags: #collaboration #notes #outline #podcast
### Goth Angel Sinners Pt. IV: The Paris Commune Notes
We're not a history podcast and nobody actually knows what happened in Paris in 1871.
January 17, 2019, tags: #collaboration #notes #outline #podcast
### A Quick Note on Monetary Theory
I’ve as of yet done little research and study on economics; in college I was well more preoccupied with physics and computer science and in my own time I’ve simply neglected it. I find this necessary to remedy, simply as a person who lives in, as they say, a society and as someone whose interests lately tend toward socioeconomics; to that end I will soon after completed a few currently in progress works be embarking...
### Goth Angel Sinners Pt. III: Spiderman 3 Notes
In this latest episode of Goth Angel Sinners, we've addressed complaints that the episodes were too long and that we are "over representing people born in 1997" while we discuss Spider-Man 3, cops, moral ambiguity, and Infinity War for some reason.
December 08, 2018, tags: #collaboration #notes #outline #podcast
### Overreliance on Connotation
First, let’s assume that all that we know is truth, or more precisely that our certainty in currently held knowledge is invariable and that said certainty for a given concept is arbitrarily high. For simplicity’s sake we will divide the set of all that is known into two smaller sets, $A$ will refer to the set of concepts that are (based on some arbitrary threshold) good and $B$ will refer to the set of all...
### Goth Angel Sinners Pt. II: Catalonia Notes
Join our hosts @trewbot and Chloe in this second installment of the Goth Angel Sinners podcast for an interview with senatorial candidate Jae Em Carico and discussions about the CNT, Revolutionary Catalonia, the Spanish Civil War, and the 2017 Catalan Independence Referendum.
September 20, 2018, tags: #collaboration #notes #outline #podcast
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2020-01-22 05:43:32
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https://www.enotes.com/homework-help/integrate-the-indefinite-integral-int-3x-3--x-2-3061767
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Integrate the indefinite integral. \int (3x^(3)-x^(2)+6x-4)/((x^(2)+1)(x^(2)+2))dx
First, we should represent the function under the integral as a sum of partial fractions. Then it will integrate easily using arctangent and/or logarithm.
The decomposition should exist in the form
( 3x^3 - x^2 + 6x - 4 ) / ( ( x^2 + 1 ) ( x^2 + 2 ) ) = ( Ax + B ) / ( x^2 + 1 ) + ( Cx + D ) / ( x^2 + 2 ) .
To find the constants, multiply by the denominator:
3x^3 - x^2 + 6x - 4 = ( Ax + B ) ( x^2 + 2 ) + ( Cx + D ) ( x^2 + 1 ) , i.e.
3x^3 - x^2 + 6x - 4 = ( A + C ) x^3 + ( B + D ) x^2 + ( 2A + C ) x + ( 2B + D ) .
This gives the system of equations
A + C = 3 , B + D = -1 , 2A + C = 6 , 2B + D = -4,
which clearly has only one solution A = 3 , C = 0 , B = -3 , D = 2 .
Then the indefinite integral is equal to
3 int x / (x^2+1) dx - 3 int (dx) / (x^2+1) dx + 2 int (dx)/(x^2+2) ,
which is simple and is equal to
3/2 ln(x^2+1) - 3 arctan (x) + sqrt(2) arctan(x/sqrt(2)) + C .
Last Reviewed by eNotes Editorial on
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2023-01-31 07:51:53
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http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=mzm&paperid=13194&option_lang=eng
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Matematicheskie Zametki
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Mat. Zametki: Year: Volume: Issue: Page: Find
Mat. Zametki, 2021, Volume 110, Issue 5, Pages 643–657 (Mi mz13194)
Categories of Symmetry Groups of the Space of Solutions of the Special Doubly Confluent Heun Equation
V. M. Buchstabera, S. I. Tertychnyib
a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
b All-Russian Scientific Research Institute of Physical-Technical and Radiotechnical Measurements, Mendeleevo, Moscow region
Abstract: The representations of the groups $G_I$, $G_II$, $G_III$, $G_IV$ that characterize symmetries of the solution space of a special doubly confluent Heun equation are described. Categories of groups whose commutator subgroup is isomorphic to the group of integers are introduced, and an algorithm for categorical characterization of such groups is described. An implementation of the algorithm for the groups $G_I$, …, $G_IV$ is given.
Keywords: doubly confluent Heun equation, symmetry groups of the space of solutions, extensions of groups, categories of groups, groups with commutator subgroup isomorphic to the group of integers.
DOI: https://doi.org/10.4213/mzm13194
Full text: PDF file (532 kB)
First page: PDF file
References: PDF file HTML file
English version:
Mathematical Notes, 2021, 110:5, 643–654
Bibliographic databases:
UDC: 512.544.42+512.583+517.923+517.926.4+512.543
Citation: V. M. Buchstaber, S. I. Tertychnyi, “Categories of Symmetry Groups of the Space of Solutions of the Special Doubly Confluent Heun Equation”, Mat. Zametki, 110:5 (2021), 643–657; Math. Notes, 110:5 (2021), 643–654
Citation in format AMSBIB
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2022-01-17 00:41:16
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https://mathoverflow.net/questions/97651/lindel%c3%b6f-subsets-of-p-spaces
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# Lindelöf subsets of $P$-spaces
A completely regular topological space $(X,\tau)$ is called a $P$-space, if every $G_\delta$-subset of $X$ is open. (i.e $\tau$ is closed under countable intersection). Here we recall some special properties of $P$-spaces:
• Every countable subset of $X$ is obviously closed and discrete.
• Every countable subset of $X$ is $C$-embedded in $X$. (i.e. every continuous real valued function on a countable subset of $X$ can be extended to all of $X$).
Now with the sake of above properties I could pose my Questions. My questions that are given as follows are the extended form of these properties of countable sets to Lindelöf subsets of $P$-spaces.
• Is it true that in every $P$-space, every Lindelöf subset is closed?
• Is it true that in every $P$-space every Lindelöf subset is $C$-embedded in $X$?
-
Dear @Hachino, please remember to not edit more than three old questions each day. (An old question here is one not already on the front page.) Thank you. You may read more at meta.mathoverflow.net/questions/599/… – Ricardo Andrade Apr 24 at 20:31
@RicardoAndrade : I hear and will comply, sorry for yesterday's burst. This shall not happen anymore. – Hachino Apr 25 at 8:33
@Hachino, no problem. It takes a while to figure out some of these "rules". Good luck! – Ricardo Andrade Apr 25 at 9:38
Every Lindelof subset of a $P$-space is closed, and the proof is almost the same as the proof of "a compact subset of a Hausdorff space is closed" (I´m assuming your space is Hausdorff since you wrote that every countable set is obviously closed).
I´m not so sure about the second question, but every $P$-space is an $F$-space and every Lindelof subset of an $F$-space is $C^*$-embedded. You could take a look at Negrepontis´ article "On the product of $F$-spaces" for a proof of this fact and try to adapt it to your situation.
Hello Dear Ramiro. At first I have to say that I am so sorry about my Delay. Thank you very much for your refrence and guidance .When I posed these problems, I wanted to improve exercise [3B] of the text gillman-jerison. For the second question, I think you were very closed to show it. It suffices to apply the following theorem for $C$-embedded subsets. Theorem:A $C^*$-embedded subset of topological space $X$ is $C$-embedded iff it is completely separated from every zero-set which is disjoint from it. As you Know in this case every zero-set is clopen and that's all. Thank's a lot. – Ali Reza May 31 '12 at 18:00
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2015-05-23 04:46:56
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https://mathshistory.st-andrews.ac.uk/HistTopics/Greek_numbers/
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# Greek number systems
There were no single Greek national standards in the first millennium BC. since the various island states prided themselves on their independence. This meant that they each had their own currency, weights and measures etc. These in turn led to small differences in the number system between different states since a major function of a number system in ancient times was to handle business transactions. However we will not go into sufficient detail in this article to examine the small differences between the system in separate states but rather we will look at its general structure. We should say immediately that the ancient Greeks had different systems for cardinal numbers and ordinal numbers so we must look carefully at what we mean by Greek number systems. Also we shall look briefly at some systems proposed by various Greek mathematicians but not widely adopted.
The first Greek number system we examine is their acrophonic system which was use in the first millennium BC. 'Acrophonic' means that the symbols for the numerals come from the first letter of the number name, so the symbol has come from an abreviation of the word which is used for the number. Here are the symbols for the numbers 5, 10, 100, 1000, 10000.
Acrophonic 5, 10, 100, 1000, 10000.
We have omitted the symbol for 'one', a simple '|', which was an obvious notation not coming from the initial letter of a number. For 5, 10, 100, 1000, 10000 there will be only one puzzle for the reader and that is the symbol for 5 which should by P if it was the first letter of Pente. However this is simply a consequence of changes to the Greek alphabet after the numerals coming from these letters had been fixed. By that time the symbols were probably not thought of as coming from the letters so there was no move to change them with changes to the symbols for the letters. The original form of π was G and so Pente was originally Gente.
Now the system was based on the additive principle in a similar way to Roman numerals. This means that 8 is simply V|||, the symbol for five followed by three symbols for one. Here is 1-10 in Greek acrophonic numbers.
1-10 in Greek acrophonic numbers.
If base 10 is used with an additive system without intermediate symbols then many characters are required to express certain numbers. The number 9999 would require 36 symbols in such a system and this is very cumbersome. We have already seen that that Greek acrophonic numbers had a special symbol for 5. This is not surprising for it cuts down the characters required and also presumably arises from counting on fingers. We have 10 fingers but there is 5 on each hand. What is slightly more surprising is that the system had intermediate symbols for 50, 500, 5000, and 50000 but they were not new characters, rather they were composite symbols made from 5 and the symbols for 10, 100, 1000, 10000 respectively. Here is how the composites were formed.
Combining acrophonic numerals.
Notice that since there was no positional aspect of the system, there was no need for zero as an empty place holder. The symbol H represented 100 as no problem is created in the representation by the number having no tens or units.
Now this is not the only way in which such composite symbols were created. We have already mentioned that different states used variants of the number system and, although we are not going to examine these in detail, let us at least give some indication by showing some forms of 50 that have been found. Most of these forms are older than the main form of the numerals we have considered being more typical of the period 1500 BC to 1000 BC.
Different forms of 50 in different Greek States.
The next point worth noting is that this number system did not really consist of abstract numbers in the way we think of numbers today. Today the number 2 is applied to any collection of two objects and 2 is thought of as an abstract property that all such collections of two objects have in common. We know that the ancient Greeks had a somewhat different idea because the numbers were used in slightly different forms depending to what the number referred. The most frequent use of this particular number system was for sums of money. The basic unit of money was the drachma with a larger unit being the talent worth 6000 drachmas. The drachma was subdivided into smaller units, namely the obol which was $\large\frac{1}{6}\normalsize$ of a drachma, and the chalkos which was $\large\frac{1}{8}\normalsize$ of an obol. Half and quarter obols were also used. Notice that this system of currency was not based on the decimal system although the number system had 10 as a base and 5 as a secondary base.
The different units of currency were denoted by modifying the notation for the units in the number.
5678 drachmas would be written in this way:
The form of the units would denote drachmas.
3807 talents would be written as:
The units would now appear as T (T for talent). A sum of money involving both drachmas and obols would be written as:
3807 drachmas and 3 obols:
This acrophonic system was used for more than money. A very similar system was also used in dealing with weights and measures which is not surprising since the value of money would certainly have evolved from a system of weights. This is confirmed by the fact that the drachma was also the name of the unit of weight.
We now look at a second ancient Greek number system, the alphabetical numerals, or as it is sometimes called, the 'learned' system. As the name 'alphabetical' suggests the numerals are based on giving values to the letters of the alphabet. It is worth noting that the Greeks were one of the first to adopt a system of writing based on an alphabet. They were not the inventors of this form of writing, for the Phoenicians had such a system in place first. The Greek alphabet used to write words was taken over from the Phoenician system and was quite close to it. We will not examine the forms of the Greek letters themselves, but it is certainly worth stressing how important this form of writing was to be in advancing knowledge. It is fundamental to our ways of communicating in most countries today, although some peoples do prefer to use other forms of writing.
There are 24 letters in the classical Greek alphabet and these were used together with 3 older letters which have fallen out of use. These 27 letters are
Of these we have given both the upper case and lower case versions of the 24 classical letters. The letters digamma, koppa, and san are the obsolete ones. Although we have not given their symbols in the above table their symbols appear in the numeral tables below. The first nine of these letters were taken as the symbols for 1, 2, ... , 9.
alphabetical 1-9.
Notice that 6 is represented by the symbol for the obsolete letter digamma.
The next nine letters were taken as the symbols for 10, 20, ... , 90.
alphabetical 10-90.
Notice that 90 is represented by the symbol for the obsolete letter koppa.
The remaining nine letters were taken as the symbols for 100, 200, ... , 900.
alphabetical 100-900.
Notice that 900 is represented by the symbol for the obsolete letter san.
Sometimes when these letters are written to represent numbers, a bar was put over the symbol to distinguish it from the corresponding letter.
Now numbers were formed by the additive principle. For example 11, 12, ... , 19 were written:
alphabetical 11-19.
Larger numbers were constructed in the same sort of way. For example here is 269.
alphabetical 269.
Now this number system is compact but without modification is has the major drawback of not allowing numbers larger than 999 to be expressed. Composite symbols were created to overcome this problem. The numbers between 1000 and 9000 were formed by adding a subscript or superscript iota to the symbols for 1 to 9.
First form of 1000, ..., 9000.
Second form of 1000, ..., 9000.
How did the Greeks represent numbers greater than 9999? Well they based their numbers larger than this on the myriad which was 10000. The symbol M with small numerals for a number up to 9999 written above it meant that the number in small numerals was multiplied by 10000. Hence writing β above the M represented 20000:
The number 20000.
Similarly written above the M represented 1230000:
The number 1230000.
Of course writing a large number above the M was rather difficult so often in such cases the small numeral number was written in front of the M rather than above it. An example from Aristarchus:
Aristarchus wrote the number 71755875 as:
For most purposes this number system could represent all the numbers which might arise in normal day to day life. In fact numbers as large as 71755875 would be unlikely to arise very often. On the other hand mathematicians did see the need to extend the number system and we now look at two such proposals, first one by Apollonius and then briefly one by Archimedes (although in fact historically Archimedes made his proposal nearly 50 years before Apollonius).
Although we do not have first hand knowledge of the proposal by Apollonius we do know of it through a report by Pappus. The system we have described above works with products by a myriad. The idea which Apollonius used to extend the system to larger numbers was to work with powers of the myriad. An M with an α above it represented 10000, M with β above it represented M2 , namely 100000000, etc. The number to be multiplied by 10000, 10000000, etc is written after the M symbol and is written between the parts of the number, a word which is best interpreted as 'plus'. As an example here is the way that Apollonius would have written 587571750269.
Apollonius's representation of 587571750269.
Archimedes designed a similar system but rather than use $10000 = 10^{4}$ as the basic number which was raised to various powers he used $100000000 = 10^{8}$ raised to powers. The first octet for Archimedes consisted of numbers up to $10^{8}$ while the second octet was the numbers from $10^{8}$ up to $10^{16}$. Using this system Archimedes calculated that the number of grains of sand which could be fitted into the universe was of the order of the eighth octet, that is of the order of 1$0^{64}$.
### References (show)
1. G Ifrah, A universal history of numbers : From prehistory to the invention of the computer (London, 1998).
Written by J J O'Connor and E F Robertson
Last Update January 2001
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2022-06-27 14:58:21
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http://red.magtech.org.cn/CN/10.13809/j.cnki.cn32-1825/te.2022.04.007
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• 方法理论 •
### 基于随机森林算法的低煤阶煤层气开发选区预测
1. 1.西安科技大学地质与环境学院,陕西 西安 710054
2.煤与煤层气共采技术国家重点实验室,山西 晋城 048000
3.陕西煤层气开发利用有限公司,陕西 西安 710065
4.自然资源部煤炭资源勘查与综合利用重点实验室,陕西 西安 710021
• 收稿日期:2022-04-12 出版日期:2022-08-26 发布日期:2022-09-02
• 通讯作者: 王丽雅 E-mail:[email protected];[email protected]
• 作者简介:陈跃（1988—）,男,博士,副教授,从事非常规天然气地质研究。地址：陕西省西安市雁塔中路58号西安科技大学,邮政编码：710054。E-mail： [email protected]
• 基金资助:
国家自然科学基金项目“润湿性制约下低阶煤不同煤岩组分甲烷解吸机制”(41902175);山西省揭榜招标项目“基于气藏工程的煤层气井人工智能排采技术与示范”(20201101002);中国博士后科学基金项目“低煤阶镜煤与暗煤润湿性差异及对甲烷解吸的影响”(2019M653873XB)
### Prediction of favorable areas for low-rank coalbed methane based on Random Forest algorithm
CHEN Yue1(),WANG Liya1(),LI Guofu2,ZHANG Lin3,YANG Fu4,MA Zhuoyuan1,GAO Zheng1
1. 1. College of Geology and Environment, Xi’an University of Science and Technology, Xi’an, Shaanxi 710054, China
2. Key Laboratory of coal and coalbed methane co-mining technology, Jincheng, Shanxi 048000, China
3. Shaanxi Coalbed Methane Development Corp.Ltd., Xi’an, Shaanxi 710065, China
4. Key Laboratory of Coal Resources Exploration and Comprehensive Utilization, Ministry of Natural Resources, Xi’an, Shaanxi 710021, China
• Received:2022-04-12 Online:2022-08-26 Published:2022-09-02
• Contact: WANG Liya E-mail:[email protected];[email protected]
Abstract:
In China, low-rank coal and coalbed methane resources are abundant, meanwhile, as a kind of clean energy, the development and utilization of coalbed methane（CBM） can effectively alleviate the shortage of natural gas resources, but the commercial scale development is slightly insufficient, and systematic research is urgently needed. The premise of efficient CBM development is the selection of favorable areas, but the current CBM development evaluation involves certain subjective human factors, which will indirectly affect or interfere with the prediction effect. Taking the low-rank coal in the Dafosi minefield in the Binchang mining area of Huanglong Coal Field as the research object, based on the actual production data, the random forest algorithm in machine learning is used to predict the favorable area of coalbed methane in the area. The results show that: ① Pearson correlation analysis shows that the gas content, ash content, net thickness of coal seam, structural position, roof thickness, permeability, reservoir pressure and burial depth are eight mutually independent CBM output-related parameters and can be used for model establishment; ② The Random Forest algorithm divides the CBM development area into five types of areas with different degrees, of which type Ⅰ（extremely high） to Ⅱ（highly favorable） areas account for 13.88 % of the entire study area, mainly distributed in the middle of the well field. The southeast is not suitable for subsequent deployment of well locations, and there is a distribution of highly favorable areas in the west, so the well locations for subsequent development and deployment should also be considered. ③ It can be obtained from the receiver operating characteristic（ROC） curve, and the area under the ROC curve （AUC） is 0.961, indicating that the Random Forest model has high prediction accuracy and reliable results. Using machine learning algorithms for comprehensive prediction of CBM favorable areas can avoid human subjective factors in traditional algorithms, and can provide a certain theoretical reference for subsequent unconventional oil and gas development and selection.
• P618.11
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2023-01-31 03:02:46
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http://www.researchgate.net/publication/227629687_The_dipole_anisotropy_of_the_2_Micron_AllSky_Redshift_Survey
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Article
# The dipole anisotropy of the 2 Micron All‐Sky Redshift Survey
Royal Observatory, Blackford Hill, Edinburgh, EH9 3HJ
(Impact Factor: 5.52). 05/2006; 368(4):1515 - 1526. DOI: 10.1111/j.1365-2966.2006.10243.x
Source: arXiv
ABSTRACT We estimate the acceleration on the Local Group (LG) from the 2 Micron All-Sky Redshift Survey (2MRS). The sample used includes about 23 200 galaxies with extinction-corrected magnitudes brighter than Ks= 11.25 and it allows us to calculate the flux-weighted dipole. The near-infrared flux-weighted dipoles are very robust because they closely approximate a mass-weighted dipole, bypassing the effects of redshift distortions and require no preferred reference frame. This is combined with the redshift information to determine the change in dipole with distance. The misalignment angle between the LG and the cosmic microwave background (CMB) dipole drops to 12°± 7° at around 50 h−1 Mpc, but then increases at larger distances, reaching 21°± 8° at around 130 h−1 Mpc. Exclusion of the galaxies Maffei 1, Maffei 2, Dwingeloo 1, IC342 and M87 brings the resultant flux dipole to 14°± 7° away from the CMB velocity dipole. In both cases, the dipole seemingly converges by 60 h−1 Mpc. Assuming convergence, the comparison of the 2MRS flux dipole and the CMB dipole provides a value for the combination of the mass density and luminosity bias parameters Ω0.6m/bL= 0.40 ± 0.09.
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• Source
##### Article: The cosmic web of the Local Universe: cosmic variance, matter content and its relation to galaxy morphology
[Hide abstract]
ABSTRACT: We present, for the first time, a characterization of the Local Universe (LU) using high precision constrained $N$-body simulations based on self-consistent phase-space reconstructions of the large-scale structure. Our study relies on the Two-Micron All-Sky Galaxy Redshift Survey. The first question we want to address is whether we live in a special cosmic web environment. We assess this problem by estimating cosmic variance from a set of unconstrained $\Lambda$CDM simulations as a function of distance to random observers. By computing volume and mass filling fractions for voids, sheets, filaments and knots, we find that the LU displays a typical scatter of about $1\sigma$ at scales $r\gtrsim15\,h^{-1}\,$Mpc, in agreement with $\Lambda$CDM, converging to a fair unbiased sample when considering spheres of about $60\,h^{-1}\,$Mpc radius. Additionally, we compute the matter density profile of the LU and found a reasonable agreement with the estimates of Karachentsev (2012) only when considering the contribution of dark haloes. This indicates that the observational estimates may be biased towards low density values. As a first application of our reconstructed cosmic web, we investigate the likelihood of different galaxy morphological types to inhabit certain cosmic web environments. In particular we find, that irrespective of the method used to define the web, either based on the density or on the peculiar velocity field, elliptical galaxies show a clear tendency to preferentially reside in clusters as opposed to voids (up to a level of $5.3\sigma$ and $9.8\sigma$ respectively) and the opposite happens to spiral galaxies (up to a level of $5.6\sigma$ and $5.4\sigma$ respectively). These findings are compatible with previous works, however providing results at higher confidence levels.
Monthly Notices of the Royal Astronomical Society 06/2014; 445(1). DOI:10.1093/mnras/stu1746 · 5.23 Impact Factor
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##### Article: Anisotropies of ultra-high energy cosmic rays diffusing from extragalactic sources
[Hide abstract]
ABSTRACT: We obtain the dipolar anisotropies in the arrival directions of ultra-high energy cosmic rays diffusing from nearby extragalactic sources. We discuss both the energy regime of spatial diffusion and the quasi-rectilinear one leading to just angular diffusion at higher energies. We obtain analytic results for the anisotropies from a single source which are validated using two different numerical simulations. For a scenario with a few sources in the local supercluster (with the closest source at a typical distance of few to tens of Mpc), we discuss the possible transition between the case in which the anisotropies are dominated by a few sources at energies below few EeV towards the regime in which many sources contribute at higher energies. The effect of a non-isotropic source distribution is also discussed, showing that it can significantly affect the observed dipole.
Physical Review D 12/2013; 89(12). DOI:10.1103/PhysRevD.89.123001 · 4.86 Impact Factor
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##### Article: Testing Isotropy in the Local Universe
[Hide abstract]
ABSTRACT: We test the isotropy of the local distribution of galaxies using the 2MASS extended source catalogue. By decomposing the full sky survey into distinct patches and using photometric redshift data, we use a combination of parametric and non-parametric statistical methods to obtain the shape and normalisation of the luminosity function in each patch. By making various magnitude and density cuts to the data, we reconstruct the luminosity density as a function of median redshift of the sample. We find evidence that the local Universe is locally underdense, with the luminosity density rising by $\sim 40\%$ between mean survey redshifts of $\bar{z} \simeq 0.03$ and $\bar{z} \simeq 0.09$. Specifically, the luminosity function normalisation rises from $\phi^{\ast} = 0.0106 \pm 0.0014 h^{3} {\rm Mpc}^{-3}$ at mean survey redshift $\bar{z} \sim 0.045$ to $\phi^{\ast} = 0.0126 \pm 0.0008 h^{3} {\rm Mpc}^{-3}$ at $\bar{z} \sim 0.09$. The data suggests that the shape of the local $K$-band luminosity function is anisotropic, exhibiting hemispherical asymmetry in the North and South Galactic plane. The inferred total luminosity density increases with survey redshift but is consistent with the assumption of isotropy over the whole sky.
Journal of Cosmology and Astroparticle Physics 05/2014; 2014(10). DOI:10.1088/1475-7516/2014/10/070 · 5.88 Impact Factor
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2015-03-30 20:32:42
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http://sciforums.com/threads/validity-of-a-simple-logical-argument.161528/page-5
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# Validity of a simple logical argument
Discussion in 'General Philosophy' started by Speakpigeon, Jan 23, 2019.
?
## Is the argument valid?
Poll closed Feb 22, 2019.
54.5%
45.5%
0 vote(s)
0.0%
0 vote(s)
0.0%
1. ### parmaleeperipatetic artisanValued Senior Member
Messages:
3,193
That's why you need a fifth option on your poll:
Is the argument valid? That depends...
But now I understand why you are going with the early 20th century, rather than, say, mid/late 19th century, so thank you for the clarification.
Messages:
1,123
EB
5. ### James RJust this guy, you know?Staff Member
Messages:
37,193
Also worth re-writing this in terms of a boolean expression. We have:
1. squid implies not giraffe.
2. giraffe implies not elephant.
3. elephant implies not squid.
4. (Joe is) squid or giraffe.
5. (Joe is) elephant.
(Conclusion. Joe is) squid.
Using the symbols "&" (conjunction), "or" (disjunction), "!" (negation), and "=>" (implication), the argument above, symbolically, is:
[(S => !G) & (G => !E) & (E => !S) & (S or G) & E] => S
Now we replace the implications, noting that (A => B) = (!A or B). The argument is equivalent to:
![(!S or !G) & (!G or !E) & (!E or !S) & (S or G) & E] or S
Continue, using various rules of Boolean algebra:
![(!S or !G) & (!G or !E) & (!E or !S) & (S or G) & E] or S
= !(!S or !G) or !(!G or !E) or !(!E or !S) or !(S or G) or !E or S
= (S & G) or (G & E) or (E & S) or (!S & !G) or !E or S
= (S & G) or (G & E) or [(E & S) or !E] or [(!S & !G) or S]
= (S & G) or (G & E) or [(E or !E) & (S or !E)] or [(!S or S) & (!G or S)]
= (S & G) or (G & E) or (S or !E) or (!G or S)
= (S & G) or (G & E) or S or !E or !G
= S or !E or (G & E) or !G
= S or !E or (G or !G) & (E or !G)
= S or !E or E or !G
= S or !G or TRUE
= TRUE
Thus we see that the given premises logically imply the given conclusion, thereby making the argument valid.
Well, that was fun. I haven't done a formal proof like this in years.
7. ### James RJust this guy, you know?Staff Member
Messages:
37,193
The problem here is that you cannot assume that all of the premises are true simultaneously.
1. A squid is not a giraffe
2. A giraffe is not an elephant
3. An elephant is not a squid
4. Joe is either a squid or a giraffe
5. Joe is an elephant
Conclusion: Therefore, Joe is a squid
Note firstly that premises (4) and (5) mandate that Joe must be a squid or a giraffe, and also that Joe must be an elephant.
If Joe is a squid, then this contradicts premises (3) and (5), taken in combination.
If Joe is a giraffe, then this contradicts premises (2) and (5), taken in combination.
If Joe is an elephant, then this contradicts premises (2) and (3) and (4), taken in combination.
Since you can't do as Aristotle directs - i.e. assume all the premises are true - then the Aristotlean validity of the argument is undecidable under Aristotle's criterion. We need a better definition of what makes a valid argument if we want every argument to be either valid or invalid, rather than having a third "undecidable" category.
Last edited: Jan 27, 2019
8. ### SpeakpigeonValued Senior Member
Messages:
1,123
I guess you understand that the notion of validity you use here is the one defined in material logic (modern mathematical "classical logic").
And so, how do you feel about the argument? Do you really feel that the conclusion follows from the premises?
EB
9. ### SpeakpigeonValued Senior Member
Messages:
1,123
Sure, the premises here could not all be true.
Yet, you're wrong when you claim here that we cannot assume that they are all true. In fact, there's absolutely no difficulty in that.
For Aristotle, premises are just stated, i.e. asserted as true, i.e. assumed true. If the conclusion then follows necessarily from the premises, the argument is valid.
Clearly, assuming all premises are true, that Joe is a Squid doesn't follow necessarily and therefore the argument is invalid.
You have to remember that the definition of validity given by Sarkus is that used for the material implication: "if, and only if, it is impossible for (all) the premises to be true and the conclusion at the same time to be false".
As such, you can't use it for validity as Aristotle described it.
In effect, the definition of validity in modern mathematical "classical logic" has been tweaked to comply with Russell's definition of the material implication. As such, it doesn't work like in Aristotle's notion of syllogism, understood as a valid argument.
Given premises 3 and 5, the conclusion not only doesn't follow but it is necessarily false.
Indeed, no conclusion based on the premises as stated could possibly be true. So, no conclusion follows at all. And I dare say, this seems exactly what sanity requires.
So, in effect, Aristotle's notion of syllogism seems to capture perfectly what we feel intuitively is a valid argument.
EB
10. ### parmaleeperipatetic artisanValued Senior Member
Messages:
3,193
My initial query arose from a misreading of your post #66, so nevermind that. Anyhows:
Your "it," in the first sentence, referring to Yazata's initial formulation in post #40 (paraphrasing, "an argument is valid iff, all premises being true, the conclusion necessarily follows").
Yeah, I can appreciate your criticisms of what is now commonly referred to as "classical logic"--well, namely that it is referred to as such. An old prof, way back when--a Russian guy who was a dead ringer for Kyle Machlachlan--pretty much shared your perspective. It kinda complicated what he was teaching, his repeated (and, at a certain stage, unnecessary ) insistence on how such and such runs counter to intuition and, well, logic; nonetheless, it was salient point.
11. ### SpeakpigeonValued Senior Member
Messages:
1,123
Aw sucks. It seemed to be a promising kerfuffle...
Not quite...
Yazata's initial version: "An argument (premises and conclusion) is valid iff, when all the premises are interpreted as being true, the conclusion must also be true."
Are interpreted as being true
... Which is always possible, and very different from "all premises being true", which is impossible in the case of this thread's Squid argument.
And that's precisely what most people fail to notice. It's not even something you will read in any logic textbook.
Thanks for sharing the anecdote. I would guess many people have some similar story. I realised there was a problem I was 19 and a maths and physics student at university. My very first time exposure to formal logic. Everything goes smoothly, conjunction, disjunction, negation etc. Arrive the implication. I stare at the truth table the guy had written on the blackboard and my brain just pukes. I can still remember this moment. So, I realised there and then there was something seriously wrong with mathematical logic. It's only later that I realised many people had a similar story to tell and indeed many philosophers have kept criticising the material implication since its inception. Without any perceptible effect. And the reason is first that the material implication is not really used to do anything (and I would certainly hope so). The second reason is that there is the Gentzen method of proof, which in effect very similar to and a generalisation of Aristotle. Seems to work well enough. The main reason is that mathematicians themselves don't even use any formal logic to prove their theorems. Meanwhile, we have this ridiculous situation where cohortes of students are taught logic as based on the material implication. This is seriously idiotic and most likely damaging in some respect.
EB
12. ### arfa branecall me arfValued Senior Member
Messages:
7,696
OMG, it's all just made-up as they go?
13. ### SpeakpigeonValued Senior Member
Messages:
1,123
Nearly all mathematicians use their logical intuitions. One or two use formal logic through theorem provers. I know of only one proof done by a mathematician using formal logic. And it's not just me who say it. One guy working on theorem provers says exactly that. And one philosopher recently wrote a paper saying the same thing. It's not exactly a secret.
EB
14. ### Write4UValued Senior Member
Messages:
18,613
Do computers use "formal" logic?
15. ### arfa branecall me arfValued Senior Member
Messages:
7,696
A question: is mathematics a formal system? If it is, does it have a formal logic?
Are there any sets of rule-based "mathematical logic"? can I just write down an intuitive equation that says 2 + 3 = 4?
16. ### SpeakpigeonValued Senior Member
Messages:
1,123
Good question. I would encourage you to start a thread on this. It's a very interesting point. I'm sure you and many people will have things to say.
EB
17. ### SpeakpigeonValued Senior Member
Messages:
1,123
Yes.
It's complicated.
There is something called "mathematical logic", but this is a field of study, not a method of logic.
There are different methods which are considered and studied in this field and are considered by mathematicians as "logics", like paraconsistent logic, relevance logic, first order logic, etc. but most of them are not what we originally mean by logic. They are best understood not as logic proper but as mathematical theories that have some similarities with logic. 1st order logic, in particular, since it is the de facto standard as far as mathematical logic goes, is at best an approximation of logic. In fact, there is an irony in the name itself, "1st order logic", because it is exactly what it is, a 1st order of approximation of logic as we think of it. As such, it has been shown to give the correct results for a number logical formulas that are basic, which is why it got selected in the first place, but you won't necessarily know if the result is still correct whenever you try with more complex formulas, and many formulas are known to give the wrong result, as exemplified in my Squid argument.
Then there is the entirely different Gentzen method of formal proof. It is used to prove mathematical theorems and produce formal proofs. However, it is still based on logical formulas, namely "rules of inference", that are themselves not proven true and therefore just admitted as true on the face of them, i.e. we all agree intuitively they must be true (e.g. "A and B implies B"). Still, as such, it is a method which is both formal and probably logical, i.e. consistent with logic as we think of it, although I couldn't possibly guaranty that.
So, Gentzem is indeed a formal method of logic for proving mathematical theorems, but it works like Aristotle's syllogism in that it is based on rules of inference admitted as true, not proven true. As far as I can tell, it is a mathematical formalisation and generalisation of Aristotle's method of formal logic.
However, very few mathematicians actually use it. It is used mainly in the context of theorem provers.
Yes, that's Gentzen.
Mathematicians start with any arbitrary axioms they want and then they try to work out all the logical consequences of those, and nearly always that will be on the basis of their own logical intuitions, not any method of formal logic.
A mathematician could start an axiomatic system with an axiom saying 2 + 3 = 4. As long as it is not logically inconsistent with another of his axioms, he will be happy. No problem with that.
However, me, I'm talking only of logical intuitions. And there's nothing logically intuitive in 2 + 3 = 4 (and there is nothing intuitive in 2+2 = 4 until you get trained in the addition). Logical intuitions are not just arbitrary ideas. A logical intuition is the subjectively certain impression that a logical relation, like for example "A and B implies A", is true. I have very good reasons to think that logical intuitions are not dependent on training in formal logic. Rather the reverse. We understand formal logic because we have logical intuitions prior to being exposed to it.
Although, I can't guarantee this applies to everybody, or even to most of us.
EB
18. ### Write4UValued Senior Member
Messages:
18,613
Is that not using a logical function based on a false premise? The premise that 5 = 4?
After all the answer is known.
Logic would demand that 2 + 3 = 5 as codified in the decimal number system (and the Fibonacci sequence), no?
19. ### James RJust this guy, you know?Staff Member
Messages:
37,193
Speaking of truth tables, here's the relevant table for A => B:
Code:
A B A=>B
0 0 1
0 1 1
1 0 0
1 1 1
If you wanted to implement implication in a computer program, say, what truth table would you prefer to use?
Write4U likes this.
Messages:
18,613
21. ### Write4UValued Senior Member
Messages:
18,613
Rather than stew over the logical validity of a proposition, we should perhaps concentrate on the real values which are presented and which will be mathematically (logically) processed;
"Joe is either a squid or a giraffe,
Joe is an elephant
Therefore, "Joe is a squid" = "Joe is an elephant" = NOT logically possible = Error.
This equation is forbidden by the physical mathematics.
Two opposing answers to an equation is mathematically impossible and therefore logically false even though the argument is logically valid until the statement of fact, a specific value, which introduces a relative constant, which now requires allowable other mathematical values of squids and giraffes, which do not exist, unless you want to go back to origins....
Try the experiment with Platonic solids if you doubt me.
Can you logically turn a sphere into a cube or an octahedron and maintain it's values and characteristics as a sphere?.............
Last edited: Jan 31, 2019
22. ### SpeakpigeonValued Senior Member
Messages:
1,123
I would want something that works.
1. So, why use this truth table?
2. Why use any truth table?
EB
23. ### SpeakpigeonValued Senior Member
Messages:
1,123
I saw one guy produce a proof that Joe is a squid. His proof is 10 lines long but I can do it 8. He infers from his proof that the argument is valid, even though you can prove using his same method that Joe is not a squid in 3 lines. Yet, he persists. He exhibited what he thinks is a proof and that's it. His method is flawed but he doesn't know that and he certainly doesn't understand why it is flawed. Using apparently the same method, I proved correctly the argument not valid:
Proof
An elephant is not a squid..............P3
Joe is an elephant...........................P5
Therefore, Joe is a not squid..........P3, P5, R1: ((x ≠ y) ∧ (z = x) ) → (z ≠ y)
Which is basically what you said.
Formally, for the argument to be valid, since equality is not a logical symbol, you need to add a premise with the R1 rule ((x ≠ y) ∧ (z = x) ) → (z ≠ y). But it's implicit anyway.
EB
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2022-08-16 22:06:38
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https://study.com/academy/answer/question-beasley-ball-bearings-paid-a-dividend-of-4-last-year-the-dividend-is-expected-to-grow-at-a-constant-rate-of-3-percent-over-the-next-five-years-the-required-rate-of-return-is-13-percent-t.html
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# Question Beasley Ball Bearings paid a dividend of $4 last year. The dividend is expected to grow... ## Question: Question Beasley Ball Bearings paid a dividend of$4 last year. The dividend is expected to grow at a constant rate of 3 percent over the next five years. The required rate of return is 13 percent (this will also serve as the discount rate in this problem).
a. Compute the anticipated value of the dividends for the next four years. (Do not round intermediate calculations. Round your final answers to 2 decimal places.) Anticipated Value:
D1 $D2$
D3 $D4$
b. Calculate the present value of each of the anticipated dividends at a discount rate of 13 percent. (Do not round intermediate calculations. Round your final answers to 2 decimal places.) PV of Dividends:
D1 $D2 D3 D4 Total$
## Constant growth in dividend:
It is assumed that dividend tends to increase over time because business firms usually grow over time. If the growth of dividend is at constant compound rate then :
Dt = D0{eq}(1 + G)^{t} {/eq}
Valuation of share where dividend increase at a constant, compound rate is given as
P0 =D1/(K-g)
K= Rate of return
g= Growth rate
a. Compute the anticipated value of the dividends for the next four years. (Do not round intermediate calculations. Round your final answers to 2 decimal places.) Anticipated Value:
D1 = D(1+g)= 4(1.03) = $4.12 D2 =D(1+g)2=4{eq}1.03^{2} {/eq} =$4.24
D3= D(1+g)3=4{eq}1.03^{3}{/eq}=$4.37 D4$ =D(1+g)4=4{eq}1.03^{4} {/eq} =$4.50 b. Calculate the present value of each of the anticipated dividends at a discount rate of 13 percent. (Do not round intermediate calculations. Round your final answers to 2 decimal places.) PV of Dividends: D1 =$4.12/1.13 = $3.65 D2 =4.24/{eq}1.13^{2} {/eq} =$3.31
D3 =4.37/{eq}1.13^{3} {/eq} =$3.01 D4 =$4.50/ ${eq}1.13^{4} {/eq} =$2.74
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2020-01-18 14:50:30
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https://repository.uantwerpen.be/link/irua/131920
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Publication
Title
Large pinning forces and matching effects in $YBa_{2}Cu_{3}O_{7-\delta}$ thin films with $Ba_{2}Y(Nb/Ta)O_{6}$ nano-precipitates
Author
Abstract
The addition of mixed double perovskite Ba2Y(Nb/Ta)O6 (BYNTO) to YBa2Cu3O7−δ (YBCO) thin films leads to a large improvement of the in-field current carrying capability. For low deposition rates, BYNTO grows as well-oriented, densely distributed nanocolumns. We achieved a pinning force density of 25 GN/m3 at 77 K at a matching field of 2.3 T, which is among the highest values reported for YBCO. The anisotropy of the critical current density shows a complex behavior whereby additional maxima are developed at field dependent angles. This is caused by a matching effect of the magnetic fields c-axis component. The exponent N of the current-voltage characteristics (inversely proportional to the creep rate S) allows the depinning mechanism to be determined. It changes from a double-kink excitation below the matching field to pinning-potential-determined creep above it.
Language
English
Source (journal)
Scientific reports. - London, 2011, currens
Publication
London : Nature Publishing Group, 2016
ISSN
2045-2322
Volume/pages
6(2016), 10 p.
Article Reference
21188
ISI
000370364500001
Medium
E-only publicatie
Full text (Publisher's DOI)
Full text (open access)
UAntwerpen
Faculty/Department Research group Publication type Subject Affiliation Publications with a UAntwerp address
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2017-06-25 08:59:59
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https://ieeexplore.ieee.org/xpl/topAccessedArticles.jsp?punumber=22
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Publication Year: 2018, Page(s):2452 - 2462
| | PDF (4434 KB) | HTML
Chipped radio-frequency identification (RFID) sensor systems have been studied for structural health monitoring (SHM) applications. However, the use of chip in sensor tags and its standardized narrowband operation contribute shortcomings in cost, durability, and detection capability. This paper presents a novel use of the frequency signature-based chipless RFID for metal crack detection and charac... View full abstract»
• ### Equivalent-circuit models for split-ring resonators and complementary split-ring resonators coupled to planar transmission lines
Publication Year: 2005, Page(s):1451 - 1461
Cited by: Papers (628) | Patents (7)
| | PDF (1031 KB) | HTML
In this paper, a new approach for the development of planar metamaterial structures is developed. For this purpose, split-ring resonators (SRRs) and complementary split-ring resonators (CSRRs) coupled to planar transmission lines are investigated. The electromagnetic behavior of these elements, as well as their coupling to the host transmission line, are studied, and analytical equivalent-circuit ... View full abstract»
• ### High-impedance electromagnetic surfaces with a forbidden frequency band
Publication Year: 1999, Page(s):2059 - 2074
Cited by: Papers (2158) | Patents (151)
| | PDF (412 KB)
A new type of metallic electromagnetic structure has been developed that is characterized by having high surface impedance. Although it is made of continuous metal, and conducts dc currents, it does not conduct ac currents within a forbidden frequency band. Unlike normal conductors, this new surface does not support propagating surface waves, and its image currents are not phase reversed. The geom... View full abstract»
• ### A Broadband Wilkinson Power Divider Based on the Segmented Structure
Publication Year: 2018, Page(s):1902 - 1911
| | PDF (2441 KB) | HTML
This paper proposes a topology of a broadband Wilkinson power divider based on the segmented structure. The segmented structure is formed by many transmission line segments in shunt with grounded capacitors and series resistor- capacitor networks. Each transmission line segment has the length of a fraction of a quarter of the wavelength (or λ/4) and the summed length of all segments remains... View full abstract»
• ### A Review of GaN on SiC High Electron-Mobility Power Transistors and MMICs
Publication Year: 2012, Page(s):1764 - 1783
Cited by: Papers (253) | Patents (6)
| | PDF (3801 KB) | HTML
Gallium-nitride power transistor (GaN HEMT) and integrated circuit technologies have matured dramatically over the last few years, and many hundreds of thousands of devices have been manufactured and fielded in applications ranging from pulsed radars and counter-IED jammers to CATV modules and fourth-generation infrastructure base-stations. GaN HEMT devices, exhibiting high power densities coupled... View full abstract»
• ### 3-D Millimeter-Wave Statistical Channel Model for 5G Wireless System Design
Publication Year: 2016, Page(s):2207 - 2225
Cited by: Papers (37)
| | PDF (4462 KB) | HTML
This paper presents a 3-D statistical channel impulse response (IR) model for urban line of sight (LOS) and non-LOS channels developed from 28- and 73-GHz ultrawideband propagation measurements in New York City, useful in the design of 5G wireless systems that will operate in both the ultra-high frequency/microwave and millimeter-wave (mmWave) spectrum to increase channel capacities. A 3GPP-like s... View full abstract»
• ### A Coupled-Line Coupling Structure for the Design of Quasi-Elliptic Bandpass Filters
Publication Year: 2018, Page(s):1921 - 1925
| | PDF (852 KB) | HTML
A coupled-line coupling structure is proposed for the design of quasi-elliptic bandpass filters (BPFs). The coupled-line structure functions as the input coupling structure for a BPF and introduces one pair of symmetrical transmission zeros at around the cutoff frequencies. Equivalent circuit models play an important role in the design of microwave filters. Accordingly, this paper undertakes a maj... View full abstract»
• ### Vertical Integration of High- $Q$ Filter With Circularly Polarized Patch Antenna With Enhanced Impedance-Axial Ratio Bandwidth
Publication Year: 2018, Page(s):3119 - 3128
| | PDF (2198 KB) | HTML
Directly coupling antennas to filters can remove the transition loss between the two and reduce the size of radio frequency front ends. Circularly polarized (CP) patch antennas with a single feed are very popular in many applications due to their compact size and ease of design. However, a probe-fed CP patch antenna typically exhibits only 1%–2% useful fractional bandwidth by ... View full abstract»
• ### Scalable Electromagnetic Energy Harvesting Using Frequency-Selective Surfaces
Publication Year: 2018, Page(s):2433 - 2441
| | PDF (2742 KB) | HTML
We present a frequency-selective surface (FSS) that is specially designed and optimized for ambient RF energy harvesting. The unit cell geometry incorporates channeling features in order to combine the collected power from multiple unit cells, allowing for efficient operation under low-power conditions. To demonstrate its performance, we designed and fabricated a matched full-wave rectifier integr... View full abstract»
• ### A 16-Element W-Band Phased-Array Transceiver Chipset With Flip-Chip PCB Integrated Antennas for Multi-Gigabit Wireless Data Links
Publication Year: 2018, Page(s):1 - 14
| | PDF (6001 KB)
This paper describes the design and implementation of a W-band phased-array system with printed circuit board (PCB) integrated antennas in two polarizations capable of multi-gigabit spectrally efficient wireless communication. The chipset is manufactured in a 0.18-μm SiGe BiCMOS technology with fT/fMAX of 240-/270-GHz and is flip-chipped onto a low-cost organic PCB with integrated antenna a... View full abstract»
• ### Design of Compact High-Isolation Four-Way Power Combiners
Publication Year: 2018, Page(s):2185 - 2198
| | PDF (4199 KB) | HTML
In this paper, the design methods for four-way power combiners based on eight-port and nine-port mode networks are proposed. The eight-port mode network is fundamentally a two-stage binary four-way power combiner composed of three magic-Ts: two compact H-plane magic-Ts and one magic-T with coplanar arms. The two compact H-plane magic-Ts and the magic-T with coplanar arms function as the first and ... View full abstract»
• ### Novel Filtering 180° Hybrid Coupler and Its Application to 2 x 4 Filtering Butler Matrix
Publication Year: 2018, Page(s):1 - 9
| | PDF (2579 KB)
In this paper, a novel filtering 180° hybrid coupler is proposed and used to design a 2 x 4 filtering Butler matrix. The filtering 180° hybrid coupler can provide power division and phase shift together with a second-order bandpass transfer function, and is based only on coupled resonators. The 2 x 4 filtering Butler matrix is easily realized by utilizing several hybrid couplers. The... View full abstract»
• ### Beam-Oriented Digital Predistortion for 5G Massive MIMO Hybrid Beamforming Transmitters
Publication Year: 2018, Page(s):1 - 14
| | PDF (3380 KB)
In this paper, we propose a beam-oriented digital predistortion (BO-DPD) technique for power amplifiers (PAs) in hybrid beamforming massive multiple-input multiple-output (MIMO) transmitters, which can achieve linearization of the transmitted signal in the main beam direction and address the DPD implementation issue in the hybrid beamforming array. In massive MIMO hybrid beamforming transmitters, ... View full abstract»
• ### Guided-wave and leakage characteristics of substrate integrated waveguide
Publication Year: 2005, Page(s):66 - 73
Cited by: Papers (519) | Patents (1)
| | PDF (432 KB) | HTML
The substrate integrated waveguide (SIW) technique makes it possible that a complete circuit including planar circuitry, transitions, and rectangular waveguides are fabricated in planar form using a standard printed circuit board or other planar processing techniques. In this paper, guided wave and modes characteristics of such an SIW periodic structure are studied in detail for the first time. A ... View full abstract»
• ### Slow-Wave Half-Mode Substrate Integrated Waveguide Using Spoof Surface Plasmon Polariton Structure
Publication Year: 2018, Page(s):2946 - 2952
| | PDF (1881 KB) | HTML
In this paper, we propose a novel slow-wave half-mode substrate integrated waveguide (HMSIW) combined with spoof surface plasmon polariton (SPP) structure. In this design, subwavelength corrugated grooves are etched on the up metal layer of HMSIW to support an SPP mode. The dispersion and transmission characteristics of the proposed hybrid HMSIW-SPP structure are analyzed and compared with the cla... View full abstract»
• ### A Novel Arbitrary Terminated Unequal Coupler With Bandwidth-Enhanced Positive and Negative Group Delay Characteristics
Publication Year: 2018, Page(s):2170 - 2184
| | PDF (3820 KB) | HTML
A novel and generalized planar coupler with four independent arbitrary terminated resistances, unequal power division, and wideband positive and negative group delays (NGDs) is proposed. The proposed asymmetrical circuit configuration includes six-section of transmission lines (TLs) and two coupled-line sections with series resistors. Six-section TLs have inherited the free impedance-transformatio... View full abstract»
• ### Adaptive Nonlinear RF Cancellation for Improved Isolation in Simultaneous Transmit–Receive Systems
Publication Year: 2018, Page(s):2299 - 2312
| | PDF (5066 KB) | HTML
This paper proposes an active radio frequency (RF) cancellation solution to suppress the transmitter (TX) passband leakage signal in radio transceivers supporting simultaneous transmission and reception. The proposed technique is based on creating an opposite-phase baseband equivalent replica of the TX leakage signal in the transceiver digital front-end through adaptive nonlinear filtering of the ... View full abstract»
• ### Wideband Bandpass Filter With Extremely Wide Upper Stopband
Publication Year: 2018, Page(s):2822 - 2827
| | PDF (1933 KB) | HTML
The realization of planar wideband bandpass filters with an extremely wide upper stopband is described. The filter design differs from conventional approaches in which it is based on the choice of a wideband bandstop (BS) filter, to which a number of shunt-shorted stubs are connected at specific nodes; the stubs provide transmission zeros at the origin. Two examples of wideband bandpass filters wi... View full abstract»
• ### New Dual-/Tri-Band Bandpass Filters and Diplexer With Large Frequency Ratio
Publication Year: 2018, Page(s):2978 - 2992
| | PDF (4581 KB) | HTML
To satisfy the simultaneous requirements of future wireless communication systems on short-range high-speed data transmission and long-distance basic coverage, the coexistence of the millimeter-wave and microwave technologies becomes the inexorable trend. However, it is difficult for the existing dual-/ tri-band filter configurations to achieve a frequency ratio larger than three. In this paper, t... View full abstract»
• ### A W-Band LNA/Phase Shifter With 5-dB NF and 24-mW Power Consumption in 32-nm CMOS SOI
Publication Year: 2018, Page(s):1973 - 1982
| | PDF (5359 KB) | HTML
This paper presents a W-band phased array receive front end in 32-nm CMOS silicon-on-insulator technology. The architecture is based on cascode low-noise amplifiers and passive switched LC 5-bit phase-shifters and with root-mean-square (rms) phase error of <;3.5° at 88-93 GHz. The 4-bit equivalent (11°) rms phase error bandwidth is 88-98 GHz. An average system noise figure (NF) of... View full abstract»
• ### Design of an 87% Fractional Bandwidth Doherty Power Amplifier Supported by a Simplified Bandwidth Estimation Method
Publication Year: 2018, Page(s):1319 - 1327
| | PDF (2576 KB) | HTML
This paper presents a novel technique for the design of broadband Doherty power amplifiers (DPAs), supported by a simplified approach for the initial bandwidth estimation that requires linear simulations only. The equivalent impedance of the Doherty inverter is determined by the value of the output capacitance of the power device, and the Doherty combiner is designed following this initial choice ... View full abstract»
• ### Design and Analysis of Schiffman Phase Shifter Under Operation of Its Second Phase Period
Publication Year: 2018, Page(s):1 - 7
| | PDF (1858 KB)
In this paper, a new design concept of Schiffman phase shifters under the operation of the second phase period in frequency domain is presented and analyzed. By investigating phase properties of the C-section structure, we first find that its second phase period is another independent and distinctive working region for Schiffman phase shifter, in addition to its first phase period. In contrast to ... View full abstract»
• ### Design Method for Multiband Filters With Compact Configuration in Substrate Integrated Waveguide
Publication Year: 2018, Page(s):3011 - 3018
| | PDF (2309 KB) | HTML
In this paper, a design method for multiband bandpass filters with compact configuration in substrate integrated waveguide is proposed by combining the single-band/dual-band synthesis method and the multimode resonator theory. The designed filters could generate multiple filtering passbands without expanding their entire physical size. In addition, all the passbands could be efficiently synthesize... View full abstract»
• ### A CMOS RF-to-DC Power Converter With 86% Efficiency and −19.2-dBm Sensitivity
Publication Year: 2018, Page(s):2409 - 2415
| | PDF (2375 KB) | HTML
This paper proposes an RF-to-dc power converter for ambient wireless powering that is efficient, highly sensitive, and less dependent on the load resistance with an extended dynamic range. The proposed rectifier utilizes a variable biasing technique to control the conduction of the rectifying transistors selectively, hence minimizing the leakage current; unlike the prior work that has a fixed feed... View full abstract»
• ### Resonant Modes of a Waveguide Iris Discontinuity: Interpretation in Terms of Canonical Circuits
Publication Year: 2018, Page(s):2059 - 2069
| | PDF (2072 KB) | HTML
This paper presents an eigenproblem for computing the resonances of multiresonant irislike discontinuities in waveguides. The formulation of the eigenproblem is based on integral equation methods, and its solution provides the resonance frequencies (eigenvalues) and the associated resonant aperture field's patterns (eigenvectors). These eigenvectors can be used as a basis for the expansion of the ... View full abstract»
• ### Constant In-Band Group-Delay Acoustic-Wave-Lumped-Element-Resonator-Based Bandpass Filters and Diplexers
Publication Year: 2018, Page(s):2199 - 2209
| | PDF (6250 KB) | HTML
The RF design of acoustic-wave (AW)-resonator-based bandpass filters (BPFs) with constant in-band group delay (τg) and analog transfer-function reconfigurability is presented. The proposed filter concept is based on N identical acoustic-wave-lumped-element resonators (AWLRs) that are electromagnetically coupled through impedance inverters and result in quasi-elliptic-type transfer functions... View full abstract»
• ### A Wideband 2$times$ 13-bit All-Digital I/Q RF-DAC
Publication Year: 2014, Page(s):732 - 752
Cited by: Papers (45) | Patents (2)
| | PDF (6002 KB) | HTML
This paper presents a wideband 2 ×13-bit in-phase/quadrature-phase (I/Q) RF digital-to-analog converter-based all-digital modulator realized in 65-nm CMOS. The isolation between I and Q paths is guaranteed employing 25% duty-cycle differential quadrature clocks. With a 1.3-V supply and an on-chip power combiner, the digital I/Q transmitter provides more than 21-dBm RF output power within a ... View full abstract»
• ### SIW Rotman Lens Antenna With Ridged Delay Lines and Reduced Footprint
Publication Year: 2018, Page(s):3136 - 3144
| | PDF (3226 KB) | HTML
A dual-layer Rotman lens with reduced footprint and wide field of view is introduced here in substrate integrated waveguide (SIW) technology in the 24-GHz band. Both layers are connected by a high-efficiency transition made by an SIW integrated reflector and several coupling vias. The delay lines of the Rotman lens are implemented in ridged waveguides for compactness and broadband operation over a... View full abstract»
• ### Highly Miniaturized 120-GHz SIMO and MIMO Radar Sensor With On-Chip Folded Dipole Antennas for Range and Angular Measurements
Publication Year: 2018, Page(s):2592 - 2603
| | PDF (5092 KB) | HTML
This paper describes a highly miniaturized two-channel radar sensor with integrated on-chip folded dipole antennas that exhibit high antenna gain and radiation efficiency due to the use of the selective localized backside etching technique. The sensor can be utilized in single-input multiple-output radar system by combining the two transmit channels to increase the effective isotropic radiated pow... View full abstract»
• ### 15 GHz Doherty Power Amplifier With RF Predistortion Linearizer in CMOS SOI
Publication Year: 2018, Page(s):1339 - 1348
| | PDF (3203 KB) | HTML
A two-stage, high-power symmetric Doherty power amplifier (PA) at 15 GHz is presented. The PA is implemented in 45 nm CMOS silicon on insulator and achieves more than 23 dB power gain with 25.7 dBm saturated output power and 31% peak power added efficiency (PAE). The 6 dB back-off PAE is 25%, which is a 64% improvement compared to ideal class B PA back-off performance. High output power is obtaine... View full abstract»
• ### Ambient RF Energy Harvesting in Urban and Semi-Urban Environments
Publication Year: 2013, Page(s):2715 - 2726
Cited by: Papers (214) | Patents (1)
| | PDF (1376 KB) | HTML
RF harvesting circuits have been demonstrated for more than 50 years, but only a few have been able to harvest energy from freely available ambient (i.e., non-dedicated) RF sources. In this paper, our objectives were to realize harvester operation at typical ambient RF power levels found within urban and semi-urban environments. To explore the potential for ambient RF energy harvesting, a city-wid... View full abstract»
• ### Substrate-Integrated Waveguide Triple-Band Bandpass Filters Using Triple-Mode Cavities
Publication Year: 2018, Page(s):2967 - 2977
| | PDF (2589 KB) | HTML
Substrate-integrated waveguide (SIW) triple-mode triple-band bandpass filters (BPFs) are synthesized, designed, and demonstrated for the first time based on a novel type of SIW triple-mode square cavities perturbed by centered cross-shaped metallized via holes. The resonant characteristics of the cavity are analyzed first to show the range of the realizable frequency ratios, and the design paramet... View full abstract»
• ### Millimeter-Wave Multifunction Multiport Interferometric Receiver for Future Wireless Systems
Publication Year: 2018, Page(s):1452 - 1466
| | PDF (3019 KB) | HTML
In this paper, a receiver architecture is presented which is capable of handling angle-of-arrival (AOA) detection as well as data communication. The architecture of the proposed multifunction receiver is based on the multiport interferometer technique, and it integrates two previously reported six-port-based system functions that were realized as two distinct six-port receivers (SPRs). This unific... View full abstract»
• ### Power amplifiers and transmitters for RF and microwave
Publication Year: 2002, Page(s):814 - 826
Cited by: Papers (683) | Patents (108)
| | PDF (365 KB) | HTML
The generation of RF/microwave power is required not only in wireless communications, but also in applications such as jamming, imaging, RF heating, and miniature dc/dc converters. Each application has its own unique requirements for frequency, bandwidth, load, power, efficiency, linearity, and cost. RF power is generated by a wide variety of techniques, implementations, and active devices. Power ... View full abstract»
• ### A Review on Recent Progress of Portable Short-Range Noncontact Microwave Radar Systems
Publication Year: 2017, Page(s):1692 - 1706
Cited by: Papers (6)
| | PDF (3657 KB) | HTML
This paper reviews recent progress of portable short-range noncontact microwave radar systems for motion detection, positioning, and imaging applications. With the continuous advancements of modern semiconductor technologies and embedded computing, many functionalities that could only be achieved by bulky radar systems in the past are now integrated into portable devices with integrated circuit ch... View full abstract»
• ### Coupling Substrate-Integrated Waveguides to Increase the Gain Bandwidth of Leaky-Wave Antennas
Publication Year: 2018, Page(s):3099 - 3109
| | PDF (3688 KB) | HTML
A novel technique to increase the pattern bandwidth of substrate-integrated waveguide leaky-wave antennas (SIW LWAs) is proposed. By coupling several SIWs, it is shown that the gain at the desired angle can be kept stable across a wide-frequency band. A systematic design methodology based on a simple transverse equivalent network is presented. Practical coupled-SIW designs with gain exceeding 10 d... View full abstract»
• ### A 6–18-GHz GaAs Multifunction Chip With 8-bit True Time Delay and 7-bit Amplitude Control
Publication Year: 2018, Page(s):2220 - 2230
| | PDF (4533 KB) | HTML
This paper presents a monolithic microwave integrated circuit multifunction chip implemented using a 0.25-μm gallium arsenide pseudomorphic high-electron-mobility transistor process for use in a phased array jammer transmitter. This chip, which operates in the frequency range of 6-18 GHz, provides several functions including 8-bit true time delay (TTD), 7-bit attenuation, wideband amplifica... View full abstract»
• ### Analytical Solutions and Characteristics for Shielded Asymmetrical Coplanar Stripline and Its Transition to Rectangular Waveguide
Publication Year: 2018, Page(s):2090 - 2099
| | PDF (1628 KB) | HTML
Analytical solutions and characteristic analysis for the shielded asymmetrical coplanar stripline (SACPS) are presented. The SACPS consists of a metal shielding cavity and an asymmetrical coplanar stripline with finite-boundary substrate. Analytical closed-form expressions for the quasi-TEM electrical parameters including the relative effective permittivity, phase velocity, and characteristic impe... View full abstract»
## Aims & Scope
The IEEE Transactions on Microwave Theory and Techniques focuses on that part of engineering and theory associated with microwave/millimeter-wave components, devices, circuits, and systems involving the generation, modulation, demodulation, control, transmission, and detection of microwave signals. This includes scientific, technical, and industrial, activities. Microwave theory and techniques relates to electromagnetic waves usually in the frequency region between a few MHz and a THz; other spectral regions and wave types are included within the scope of the Society whenever basic microwave theory and techniques can yield useful results. Generally, this occurs in the theory of wave propagation in structures with dimensions comparable to a wavelength, and in the related techniques for analysis and design..
Full Aims & Scope
## Meet Our Editors
Editor-in-Chief
Luca Perregrini
[email protected]
Editor-in-Chief
Jose Carlos Pedro
[email protected]
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2018-06-21 10:43:58
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https://brilliant.org/discussions/thread/iit-jee-2017/
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×
# IIT JEE 2017
This is discussion forum for all JEE aspirants. Pls spread this message to enable more students to be part of this group. Thanks.
Note by Ritu Roy
8 months ago
Sort by:
I am appearing in JEE 2017 · 8 months ago
Great. Same here. Do you attend coaching for the same · 8 months ago
Yeah FIITJEE! · 7 months, 4 weeks ago
Anybody nervous about the jee?????? · 6 months ago
So, Who all are appearing in jee mains 2017 ? · 8 months ago
Are you on the list? · 8 months ago
Yes. · 8 months ago
I'm also appearing for JEE 2017 · 4 months, 1 week ago
I'm also appearing for Mains 2017 and Advanced 2017 · 5 months, 2 weeks ago
I m appearing in 2017 · 7 months, 4 weeks ago
I am appearing in JEE -17. I Go to Resonance. · 8 months ago
I use resonance dlpd resources. · 8 months ago
I Am Appearing in JEE 2017 . Yeah i go to fiitjee · 8 months ago
And i attend Brilliant Coaching. Did your centre start 12th grade portions? · 8 months ago
Well we are on Functions in maths,sound waves in physics and aldehyde ketone in chemistry · 8 months ago
Oh..We are still in the 11th portions. So you must have completed a lot in organic · 7 months, 3 weeks ago
Yep we are towards the end of organic · 7 months, 3 weeks ago
me too appearing for jee 2017 . · 7 months, 3 weeks ago
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2016-10-01 22:22:11
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https://homework.cpm.org/category/CCI_CT/textbook/int2/chapter/8/lesson/8.1.1/problem/8-20
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### Home > INT2 > Chapter 8 > Lesson 8.1.1 > Problem8-20
8-20.
Marty and Gerri are playing Pick a Tile, a game in which the player reaches into two bags. One bag contains square tiles and the other circular tiles. The bag with squares contains three yellow, one blue, and two red squares. The bag with circles has one yellow and two red circles. In order to win the game (and a large stuffed animal), a player must choose one blue square and one red circle.
1. Complete the two-way table below.
2. What is the probability of a player choosing the winning blue-red combination?
1. When Marty pulls her hand out of the bag, Gerri squeals with delight because she thinks she sees something blue. If it is something blue, what is the probability that Marty won a stuffed animal?
$\approx66.7\%$
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2020-05-28 03:40:42
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https://tex.stackexchange.com/questions/418950/problems-with-colourful-exercise-environment-in-latex
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# Problems with colourful exercise environment in latex
I create an exercise sheet with solutions and I would like to use the following colourful exercise environment that I also found on this blog :
\documentclass[a4paper,10pt,oneside]{article}
\usepackage[paper=a4paper,left=35mm,right=20mm,top=25mm,bottom=20mm]{geometry}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[english]{babel}
\usepackage{nth}
\usepackage{xspace}
\usepackage{amsmath}
\usepackage{pdfpages}
\usepackage{amssymb}
\usepackage{amstext}
\usepackage{amsthm}
\usepackage{amsfonts}
\usepackage{graphicx}
\usepackage{paralist}
\usepackage{acronym}
\usepackage{xcolor}
\usepackage{enumitem}
\usepackage{fancyhdr}
\usepackage{tikz}
\usepackage{tikz-cd}
\usepackage{mathrsfs}
\usepackage{float}
\usepackage{subfigure}
\usepackage{leftidx}
\usepackage{lipsum}
\usepackage[many]{tcolorbox}
\usetikzlibrary{matrix,arrows,decorations.pathmorphing}
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\xp}{x^{\prime}}
\let \ra \rangle
\let \l \langle
\newcommand{\oco}{\otimes \cdots \otimes}
\newcommand{\dos}{, \ldots ,}
\newcommand{\tis}{ \t \cdots \t}
\newcommand{\bt}[1]{\left| #1 \right|}
\newcommand{\pr}[1]{#1^{\prime}}
\newcommand{\fk}[1]{\textit{\textbf{#1}}}
\newcommand{\h}{^} % I use this command instead of ^ since my notebook has a different keyboard on which it is cumbersome to find the sign ^
\newcommand{\tr}[1]{\textit{\textcolor{red}{#1}}}
\let\t\times
\let\r\rightarrow
\newtheorem{theorem}{Theorem}
\newtheorem{corollary}{Corollary}[theorem]
\newtheorem{lemma}[theorem]{Lemma}
\theoremstyle{definition}
\newtheorem{definition}{Definition}
\theoremstyle{definition}
\newtheorem{prop}{Proposition}
\theoremstyle{definition}
\newtheorem*{exmps}{Examples}
\theoremstyle{definition}
\newtheorem*{exmpn}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}[section]
\theoremstyle{definition}
\newtheorem{rems}{Remarks}[section]
\theoremstyle{definition}
\newtheorem{exmp}{Example}[section]
\definecolor{greentitle}{rgb}{0.0, 0.0, 0.5}
\definecolor{greentitleback}{rgb}{0.8, 0.8, 1.0}
\newtcolorbox[
auto counter,
number within=section
]{exo}[2][]{%
breakable,
enhanced,
colback=white,
colbacktitle=white,
arc=0pt,
leftrule=1pt,
rightrule=0pt,
toprule=0pt,
bottomrule=1pt,
titlerule=0pt,
colframe=greentitleback,
fonttitle=\normalcolor,
overlay={
\node[
outer sep=0pt,
anchor=east,
text width=2.5cm,
minimum height=4ex,
fill=greentitleback,
font=\color{greentitle}\sffamily\scshape
] at (title.west) {exercise~\thetcbcounter};
},
title=#2,
#1
}
\newcommand\Solution{\par\textbf{\textit{\textcolor{greentitle}{Solution :}}}\par\medskip}
\begin{document}
\section{Exercises on categories, functors and natural transformations}
\ \\
\begin{exo}{Let $\mathcal{C}$ and $\mathcal{D}$ be two categories.}
\begin{enumerate}
\item[a)] Prove that $\mathcal{C}\h{op}$ is a category, called the \tr{opposite category}.
\item[b)] Show that $\mathcal{C} \t \mathcal{D}$ is a category, termed the \tr{product category}.
\end{enumerate}
\Solution
\begin{enumerate}
\item[a)] Let us remind that the opposite category $\mathcal{C}\h{op}$ of a given category $\mathcal{C}$ is obtained by reversing the morphisms. Clearly, we have $(\mathcal{C}\h{op})\h{op} = \mathcal{C}$, since reversing the arrows twice yields the original category. It is easily comprehensible that $\mathcal{C}\h{op}$ is formed as follows :
\begin{enumerate}
\item[\textcolor{greentitle}{•}] The objects of $\mathcal{C}\h{op}$ coincide with the objects of $\mathcal{C}$, i.e. $Ob(\mathcal{C}\h{op}) = Ob(\mathcal{C})$.
\item[\textcolor{greentitle}{•}] Let $A, B \in Ob(\mathcal{C})$. To every morphism $f : A \r B$ in $C$, there corresponds a morphism $f\h{op} : B \r A$ in $C\h{op}$. In other words, $Hom_{\mathcal{C}}(A, B) = Hom_{\mathcal{C}\h{op}}(B, A)$.
\item[\textcolor{greentitle}{•}] If $f: A \r B$ and $g: B \r C$ are morphisms in $\mathcal{C}$, then the composite
\ \\ $(g \circ f)\h{op} = f\h{op} \circ g\h{op}$ in $\mathcal{C}\h{op}$ is defined to be the composite $g \circ f$ in $\mathcal{C}$, as illustrated in the diagram below.
\begin{figure}[H]
\centerline{\includegraphics[width=7cm]{d1col.png}}
\end{figure}
\ \\ Since $\mathcal{C}$ is a category, it is evident that the composition in $\mathcal{C}\h{op}$ is associative.
\item[\textcolor{greentitle}{•}] The identity of $\mathcal{C}\h{op}$ is the equivalent to the identity of $\mathcal{C}$.
\end{enumerate}
\ \\ We conclude that $\mathcal{C}\h{op}$ is a category.
\item[b)] The product category $\mathcal{C} \t \mathcal{D}$ is composed of the following :
\begin{enumerate}
\item[\textcolor{greentitle}{•}] The objects of $\mathcal{C} \t \mathcal{D}$ are pairs of objects $(A, B)$, where $A \in Ob(\mathcal{C})$ and $B \in Ob(\mathcal{D})$.
\item[\textcolor{greentitle}{•}] The morphisms from $(A, B)$ to $(\pr{A}, \pr{B})$ are pairs of morphisms $(f, g)$,
\ \\ where $f : A \r \pr{A}$ is a morphism of $\mathcal{C}$ and $g : B \r \pr{B}$ is a morphism of $\mathcal{D}$.
\item[\textcolor{greentitle}{•}] The composition of two morphisms $(f, g) : (A, B) \r (\pr{A}, \pr{B})$ and
\ \\ $(\pr{f}, \pr{g}) : (\pr{A}, \pr{C}) \r (A\h{\prime \prime}, B\h{\prime \prime})$ in $\mathcal{C} \t \mathcal{D}$ is defined componentwise by means of the composites in $\mathcal{C}$ and $\mathcal{D}$, i.e. $(\pr{f}, \pr{g}) \circ (f, g) = (\pr{f} \circ f, \pr{g} \circ g)$. Moreover, associativity of the composition of morphisms in $\mathcal{C} \t \mathcal{D}$ follows from the fact that the composition of morphisms in the categories $\mathcal{C}$ and $\mathcal{D}$ are associative.
\item[\textcolor{greentitle}{•}] The identity of $\mathcal{C} \t \mathcal{D}$ is the pair $1_{(A, B)} = (1_A, 1_B)$ of identities of the categories $\mathcal{C}$ and $\mathcal{D}$ respectively.
\end{enumerate}
\end{enumerate}
\end{exo}
\ \\
\begin{exo}{Recall the definition of a cochain complex of vector spaces from the lecture. Show that $cCh(Vect)$ and $gVect$ are categories.} Subsequently, prove that $H : cCh(Vect) \r gVect$ is a functor, called the \tr{cohomology functor}.
\ \\
\Solution
\ \\ (The solution is not complete.)It is straightforward to show that $gVect$ is a category : Let $K$ be a field. The objects of $gVect$ are $K$-vector spaces $V$ equipped with a direct sum decomposition $V = \bigoplus\limits_{k \in \Z} V_k$.
\ \\ Let $U, V \in Ob(gVect)$. Morphisms of $\Z$-graded vector spaces are $K$-linear maps $f: U \r V$ of degree $0$, i.e. $f(U_i) \subseteq V_i$ for all $i \in \Z$. For every $U, V$ and $W \in Ob(gVect)$, the composition of morphisms $f: U \r V$ and $g : V \r W$ is defined as $g \circ f : U \r W$. The identity of $gVect$ is given by the morphism $id_V : V \r V$.
\end{exo}
\begin{exo}{Verify that the map $T : Manifold \r VectBd$ sending a smooth manifold $M$ to its tangent bundle $TM$ and a smooth map $f: M \r N$ to its derivative $Tf : TM \r TN$ is a covariant functor from the category of smooth manifolds to the category of vector bundles. }
\Solution
\end{exo}
\begin{exo}{Show that functors preserve isomorphisms.}
\ \\
\Solution
\end{exo}
\end{document}
However, I have two problems. Each time a new exercise begins, its numbering is indicated by a coloured box on the left hand side. The first problem is that, if an exercise with solution continues over several pages, then the coloured box indicating the numbering of the exercise also appears several times, but it should only appear once at the beginning of the new exercise.
The second problem is that I cannot write commas in the title of an exercise, because otherwise the pgfkeys error message occurs.
I would be very pleased if someone can help me to solve my problems. Unfortunately, I am not very experienced in using latex.
• Welcome to TeX.SX! Please provide a compilable document, not just a fragment. – Bobyandbob Mar 7 '18 at 17:41
• It would help to have a compilable document and not fragments. And since your problems are related to tcolorbox, looking into the manual of that package could also help, especially for overlay features – user31729 Mar 7 '18 at 18:00
• Ok, I posted the complete code. There are also some commands inside that are not necessary. – Crystal Mar 7 '18 at 18:13
• you upload complete document :-(, but we only need a minimal work example (mwe) which exhibits your problem. for real text use dummy text generated by package lipsum or other similar. we are not so familiar with your document as you are, so we can go lost in it ... and real image replace with example-image from graphicx package. please, remove all what is not necessary for mwe – Zarko Mar 7 '18 at 18:20
• This is neither blog nor a forum ... – user31729 Mar 7 '18 at 19:08
The overlay unbroken and first option will use the overlay specifications either if the box is unbroken (i.e. for short boxes) or only for the first part of the broken box.
The , issue is due to the fact, that it should be title={#2}, not title=#2,
There are so many other issues here regarding style etc, but that are not part of the question and should not made part of the question.
\documentclass[a4paper,10pt,oneside,demo]{article}
\usepackage[paper=a4paper,left=35mm,right=20mm,top=25mm,bottom=20mm]{geometry}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[english]{babel}
\usepackage{nth}
\usepackage{xspace}
\usepackage{amsmath}
\usepackage{pdfpages}
\usepackage{amssymb}
\usepackage{amstext}
\usepackage{amsthm}
\usepackage{amsfonts}
%\PassOptionsToPackage{demo}{graphicx}
\usepackage{graphicx}
\usepackage{paralist}
\usepackage{acronym}
\usepackage{xcolor}
\usepackage{enumitem}
\usepackage{fancyhdr}
\usepackage{tikz}
\usepackage{tikz-cd}
\usepackage{mathrsfs}
\usepackage{float}
\usepackage{subfigure}
\usepackage{leftidx}
\usepackage{lipsum}
\usepackage[many]{tcolorbox}
\usepackage{blindtext}
\usetikzlibrary{matrix,arrows,decorations.pathmorphing}
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\xp}{x^{\prime}}
\let \ra \rangle
\let \l \langle
\newcommand{\oco}{\otimes \cdots \otimes}
\newcommand{\dos}{, \ldots ,}
\newcommand{\tis}{ \t \cdots \t}
\newcommand{\bt}[1]{\left| #1 \right|}
\newcommand{\pr}[1]{#1^{\prime}}
\newcommand{\fk}[1]{\textit{\textbf{#1}}}
\newcommand{\h}{^} % I use this command instead of ^ since my notebook has a different keyboard on which it is cumbersome to find the sign ^
\newcommand{\tr}[1]{\textit{\textcolor{red}{#1}}}
\let\t\times
\let\r\rightarrow
\newtheorem{theorem}{Theorem}
\newtheorem{corollary}{Corollary}[theorem]
\newtheorem{lemma}[theorem]{Lemma}
\theoremstyle{definition}
\newtheorem{definition}{Definition}
\theoremstyle{definition}
\newtheorem{prop}{Proposition}
\theoremstyle{definition}
\newtheorem*{exmps}{Examples}
\theoremstyle{definition}
\newtheorem*{exmpn}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}[section]
\theoremstyle{definition}
\newtheorem{rems}{Remarks}[section]
\theoremstyle{definition}
\newtheorem{exmp}{Example}[section]
\definecolor{greentitle}{rgb}{0.0, 0.0, 0.5}
\definecolor{greentitleback}{rgb}{0.8, 0.8, 1.0}
\newtcolorbox[
auto counter,
number within=section
]{exo}[2][]{%
breakable,
enhanced,
colback=white,
colbacktitle=white,
arc=0pt,
leftrule=1pt,
rightrule=0pt,
toprule=0pt,
bottomrule=1pt,
titlerule=0pt,
colframe=greentitleback,
fonttitle=\normalcolor,
overlay unbroken and first={
\node[
outer sep=0pt,
anchor=east,
text width=2.5cm,
minimum height=4ex,
fill=greentitleback,
font=\color{greentitle}\sffamily\scshape
] at (title.west) {exercise~\thetcbcounter};
},
title={#2},
#1
}
\newcommand\Solution{\par\textbf{\textit{\textcolor{greentitle}{Solution :}}}\par\medskip}
\begin{document}
\section{Exercises on categories, functors and natural transformations}
\ \\
\begin{exo}{Let $\mathcal{C}$, and $\mathcal{D}$ be two categories.}% comma explicitly there
\begin{enumerate}[label={\alph*)}]
\item Prove that $\mathcal{C}\h{op}$ is a category, called the \tr{opposite category}.
\item Show that $\mathcal{C} \t \mathcal{D}$ is a category, termed the \tr{product category}.
\end{enumerate}
\Solution
\begin{enumerate}
\item[a)] Let us remind that the opposite category $\mathcal{C}\h{op}$ of a given category $\mathcal{C}$ is obtained by reversing the morphisms. Clearly, we have $(\mathcal{C}\h{op})\h{op} = \mathcal{C}$, since reversing the arrows twice yields the original category. It is easily comprehensible that $\mathcal{C}\h{op}$ is formed as follows :
\begin{enumerate}
\item[\textcolor{greentitle}{•}] The objects of $\mathcal{C}\h{op}$ coincide with the objects of $\mathcal{C}$, i.e. $Ob(\mathcal{C}\h{op}) = Ob(\mathcal{C})$.
\item[\textcolor{greentitle}{•}] Let $A, B \in Ob(\mathcal{C})$. To every morphism $f : A \r B$ in $C$, there corresponds a morphism $f\h{op} : B \r A$ in $C\h{op}$. In other words, $Hom_{\mathcal{C}}(A, B) = Hom_{\mathcal{C}\h{op}}(B, A)$.
\item[\textcolor{greentitle}{•}] If $f: A \r B$ and $g: B \r C$ are morphisms in $\mathcal{C}$, then the composite
\ \\ $(g \circ f)\h{op} = f\h{op} \circ g\h{op}$ in $\mathcal{C}\h{op}$ is defined to be the composite $g \circ f$ in $\mathcal{C}$, as illustrated in the diagram below.
\begin{figure}[H]
\centering
\includegraphics[width=7cm]{d1col.png}
\end{figure}
\ \\ Since $\mathcal{C}$ is a category, it is evident that the composition in $\mathcal{C}\h{op}$ is associative.
\item[\textcolor{greentitle}{•}] The identity of $\mathcal{C}\h{op}$ is the equivalent to the identity of $\mathcal{C}$.
\end{enumerate}
\ \\ We conclude that $\mathcal{C}\h{op}$ is a category.
\item[b)] The product category $\mathcal{C} \t \mathcal{D}$ is composed of the following :
\begin{enumerate}
\item[\textcolor{greentitle}{•}] The objects of $\mathcal{C} \t \mathcal{D}$ are pairs of objects $(A, B)$, where $A \in Ob(\mathcal{C})$ and $B \in Ob(\mathcal{D})$.
\item[\textcolor{greentitle}{•}] The morphisms from $(A, B)$ to $(\pr{A}, \pr{B})$ are pairs of morphisms $(f, g)$,
\ \\ where $f : A \r \pr{A}$ is a morphism of $\mathcal{C}$ and $g : B \r \pr{B}$ is a morphism of $\mathcal{D}$.
\item[\textcolor{greentitle}{•}] The composition of two morphisms $(f, g) : (A, B) \r (\pr{A}, \pr{B})$ and
\ \\ $(\pr{f}, \pr{g}) : (\pr{A}, \pr{C}) \r (A\h{\prime \prime}, B\h{\prime \prime})$ in $\mathcal{C} \t \mathcal{D}$ is defined componentwise by means of the composites in $\mathcal{C}$ and $\mathcal{D}$, i.e. $(\pr{f}, \pr{g}) \circ (f, g) = (\pr{f} \circ f, \pr{g} \circ g)$. Moreover, associativity of the composition of morphisms in $\mathcal{C} \t \mathcal{D}$ follows from the fact that the composition of morphisms in the categories $\mathcal{C}$ and $\mathcal{D}$ are associative.
\item[\textcolor{greentitle}{•}] The identity of $\mathcal{C} \t \mathcal{D}$ is the pair $1_{(A, B)} = (1_A, 1_B)$ of identities of the categories $\mathcal{C}$ and $\mathcal{D}$ respectively.
\end{enumerate}
\end{enumerate}
\end{exo}
\ \\
\begin{exo}{Recall the definition of a cochain complex of vector spaces from the lecture. Show that $cCh(Vect)$ and $gVect$ are categories.} Subsequently, prove that $H : cCh(Vect) \r gVect$ is a functor, called the \tr{cohomology functor}.
\ \\
\Solution
\ \\ (The solution is not complete.)It is straightforward to show that $gVect$ is a category : Let $K$ be a field. The objects of $gVect$ are $K$-vector spaces $V$ equipped with a direct sum decomposition $V = \bigoplus\limits_{k \in \Z} V_k$.
\ \\ Let $U, V \in Ob(gVect)$. Morphisms of $\Z$-graded vector spaces are $K$-linear maps $f: U \r V$ of degree $0$, i.e. $f(U_i) \subseteq V_i$ for all $i \in \Z$. For every $U, V$ and $W \in Ob(gVect)$, the composition of morphisms $f: U \r V$ and $g : V \r W$ is defined as $g \circ f : U \r W$. The identity of $gVect$ is given by the morphism $id_V : V \r V$.
\end{exo}
\begin{exo}{Verify that the map $T : Manifold \r VectBd$ sending a smooth manifold $M$ to its tangent bundle $TM$ and a smooth map $f: M \r N$ to its derivative $Tf : TM \r TN$ is a covariant functor from the category of smooth manifolds to the category of vector bundles. }
\Solution
\end{exo}
\begin{exo}{Show that functors preserve isomorphisms.}
\blindtext[20]
\Solution
\end{exo}
\end{document}
• Thank you very much for helping me, although my code is a bit chaotic. I think you solved my problems. I still have a little question : The document now indicates a black box instead of my original picture. How can I change this again ? – Crystal Mar 7 '18 at 19:43
• Remove the demo option from the documentclass -- that was one possibility to make your document compilable – user31729 Mar 7 '18 at 19:45
|
2020-04-01 21:43:49
|
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|
http://crypto.stackexchange.com/questions/8893/how-does-partial-key-escrow-work/8894
|
# How does partial key escrow work?
It is being speculated that one of the ways that NSA leaker Edward Snowden may have created his "insurance policy" – distributing sensitive documents to various individuals with instructions to disseminate them should anything happen to him--is through the use of a partial key escrow.
The concept is described vaguely in the article:
Modern crypto has lots of ways to accomplish this, including partial key escrow, in which a bunch of encryption keys are distributed to various parties; the individual keys aren't sufficient to decrypt the file, but a quorum of them are. For example, you can release a key to ten people and configure it so that any three can, in combination, unlock the file.
From a technical standpoint, how does partial key escrow work, and how is it most often implemented in practice?
-
## migrated from security.stackexchange.comJun 27 '13 at 12:44
This question came from our site for Information security professionals.
Shamir's Secret Sharing Scheme... (Not a crypto geek, but I suspect there are other similar concepts). – Deer Hunter Jun 26 '13 at 6:12
Another note: speculations of that kind are mostly sorry fruit of barren tabloid minds.. – Deer Hunter Jun 26 '13 at 6:14
Interesting question. And if someone gets 3 keys, would she need to use them in a specific order and provide passwords? – Aki Jun 26 '13 at 8:18
@Aki - no. I suppose you are thinking of rubber-hose cryptography, then. – Deer Hunter Jun 26 '13 at 9:46
I couldn't even find information other than theory about this "partial key escrow". My bet is the truecrypt-style keyfiles where you need all the "keys" and a password to decrypt the data. – Nathan C Jun 27 '13 at 2:25
The two most popular ways I am aware of are Shamir secret sharing and additive secret sharing. I'll explain both.
I'll start with additive as it is conceptually simpler (but also more limited). I'll also use bitwise addition modulo 2 as the addition operation (i.e., XOR), but know that that isn't the only option. You could use real, no-kidding addition in a finite field (say $\mathbb{Z}_p$ for a prime $p$).
Say I have a key $k$ and a plaintext $p$. I encrypt to get $c=E(p,k)$. Now I distribute $c$ to $n$ friends. I also generate random bitstrings of the same length as $k$, $s_1,\dots,s_{n-1}$. I also set $s_n=k\oplus s_1\oplus\dots\oplus s_{n-1}$. I send $s_i$ to friend $i$. Note that $k=s_1\oplus\dots\oplus s_{n}$. Thus, if all $n$ friends get together, they can reconstruct $k$ and decrypt $c$. If any group of up to $n-1$ of my friends get together they can not reconstruct $k$ and they learn no additional information about $k$ that they didn't already have.
Shamir secret sharing is more complicated, but more powerful. In SSS, I can generate the shares such that any $t+1$ out of $n$ of my friends can get together to reconstruct $k$. Any group of at most size $t$ learns nothing about $k$. It is based on the idea of polynomial interpolation. For example, in school we learned that it takes 2 points to uniquely identify a line. If you only have one point, there are infinitely many lines that could touch that point. If you have 2 points, there is only 1 line. So, what if we encoded the secret $k$ into the line (for example, the y intercept). Then I can distribute different points on that line (excluding the y intercept) to all my friends. Any 1 of them by themselves, cannot figure out the y intercept. But, 2 of them together can reconstruct the line and figure out the y intercept and recover $k$.
SSS uses this idea. Fix a field, say $\mathbb{Z}_p$. Construct the following polynomial:
$f(x)=k+r_1x+r_2x^2+\dots+r_tx^t$ where $r_1,\dots,r_t$ are random field elements. Notice that $f(0)=k$. Send $f(1)$ to one friend, $f(2)$ to another, and so on.
It turns out that any $t+1$ of your friends can then come together and, using lagrangian interpolation, compute $f(0)=k$ to recover the secret $k$. Any group of size at most $t$, however, learns nothing about $k$.
|
2015-04-27 10:42:33
|
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|
https://en.wikibooks.org/wiki/Introduction_to_Chemical_Engineering_Processes/Steady_state_energy_balance
|
# Introduction to Chemical Engineering Processes/Steady state energy balance
## General Balance Equation Revisited
Recall the general balance equation that was derived for any system property:
${\displaystyle In-Out+Generation-Consumption=Accumulation}$
When we derived the mass balance, we did so by citing the law of conservation of mass, which states that the total generation of mass is 0, and therefore ${\displaystyle Accumulation=In-Out}$.
There is one other major conservation law which provides an additional equation we can use: the law of conservation of energy. This states that if E denotes the entire amount of energy in the system,
Law of Conservation of Energy ${\displaystyle E_{in}-E_{out}=E_{accumulated}}$
## Types of Energy
In order to write an energy balance, we need to know what kinds of energy can enter or leave a system. Here are some examples (this is not an exhaustive list by any means) of the types of energy that can be gained or lost.
1. A system could gain or lose kinetic energy, if we're analyzing a moving system.
2. Again, if the system is moving, there could be potential energy changes.
3. Heat could enter the system via conduction, convection, or radiation.
4. Work (either expansion work or shaft work) could be done on, or by, the system.
The total amount of energy entering the system is the sum of all of the different types entering the system. Here are the expressions for the different types of energy:
1. From physics, recall that ${\displaystyle KE={\frac {1}{2}}mv^{2}}$. If the system itself is not moving, this is zero.
2. The gravitational potential energy of a system is ${\displaystyle GPE=mgh}$ where g is the gravitational constant, m is mass in kg and h is the height of the center of mass of the system. If the system does not change height, there is no change in GPE.
3. The heat entering the system is denoted by Q, regardless of the mechanism by which it enters (the means of calculating this will be discussed in a course on transport phenomenon). According to this book's conventions, heat entering a system is positive and heat leaving a system is negative, because the system in effect gains energy when heat enters.
4. The work done by or on the system is denoted by W. Work done BY a system is negative because the system has to "give up" energy to do work on its surroundings. For example, if a system expands, it loses energy to account for that expansion. Conversely, work done ON a system is positve.
## Energy Flows due to Mass Flows
Accumulation of anything is 0 at steady state, and energy is no exception. If, as we have the entire time, we assume that the system is at steady state, we obtain the energy balance equation:
${\displaystyle E_{in}=E_{out}}$
This is the starting point for all of the energy balances below.
Consider a system in which a mass, such as water, enters a system, such as a cup, like so:
The mass flow into (or out of) the system carries a certain amount of energy, associated with how fast it is moving (kinetic energy), how high off the ground it is (potential energy), and its temperature (internal energy). It is possible for it to have other types of energy as well, but for now let's assume that these are the only three types of energy that are important. If this is true, then we can say that the total energy carried in the flow itself is:
${\displaystyle {\dot {E}}_{i}=({\frac {1}{2}}{\dot {m}}v^{2}+{\dot {m}}gh+{\dot {U}})_{i}}$
However, there is one additional factor that must be taken into account. When a mass stream flows into a system it expands or contracts and therefore performs work on the system. An expression for work due to this expansion is:
${\displaystyle W_{exp}=P*{\dot {V}}_{i}}$
Since this work is done on the system, it enters the energy balance as a positive quantity. Therefore the total energy flow into the system due to mass flow is as follows:
${\displaystyle {\dot {E}}_{i}=({\frac {1}{2}}{\dot {m}}v^{2}+{\dot {m}}gh+{\dot {U}})_{i}+P*{\dot {V}}_{i}}$
Now, to simplify the math a little bit, we generally don't use internal energy and the PV term. Instead, we combine these terms and call the result the enthalpy of the stream. Enthalpy is just the combination of internal energy and expansion work due to the stream's flow, and is denoted by the letter H:
Definition of enthalpy ${\displaystyle H=U+PV}$
Therefore, we obtain the following important equation for energy flow carried by mass:
In stream i, if only KE, GPE, internal energy, and expansion work are considered, the energy carried by mass flow is: ${\displaystyle {\dot {E}}_{i}=({\frac {1}{2}}{\dot {m}}v^{2}+{\dot {m}}gh+{\dot {H}})_{i}}$
Note: Kinetic energy and potential energy are generally very small compared to the enthalpy, except in cases of very rapid flow or when there are no significant temperature changes occurring in the system. Therefore, they are often neglected when performing energy balances.
## Other energy flows into and out of the system
The other types of energy flows that could occur in and out of a system are heat and work. Heat is defined as energy flow due to a change in temperature, and always flows from higher temperature to lower temperature. Work is defined as an energy transferred by a force (see here for details).
• If there is no heat flow into or out of a system, it is referred to as adiabatic.
• If there are no mechanical parts connected to a system, and the system is not able to expand, then the work is essentially 0.
Some systems which have mechanical parts that perform work are turbines, mixers, engines, stirred tank reactors, agitators, and many others. The type of work performed by these parts is called shaft work to distinguish it from work due to expansion of the system itself (which is called expansion work).
An "insulated system" is generally interpreted as being essentially adiabatic, though how good this assumption is depends on the quality of the insulation. A system that cannot expand is sometimes described as "rigid".
The notation for these values are as follows:
• Heat flows: ${\displaystyle {\dot {Q}}_{j}}$, at the "j"th location.
• Shaft work: ${\displaystyle {\dot {W}}_{s}}$
• Expansion work: ${\displaystyle P*{\frac {\Delta {V}}{\Delta {t}}}}$
Note that the above implies that there is no expansion work at steady state because at steady state nothing about the system, including the volume, changes with time, i.e. ${\displaystyle {\frac {\Delta {V}}{\Delta {t}}}=0{\mbox{ at steady state}}}$.
## Overall steady - state energy balance
If we combine all of these components together, remembering that heat flow into a system and work done on a system are positive, we obtain the following:
Steady State Energy Balance on an Open System ${\displaystyle \Sigma ({\frac {1}{2}}{\dot {m}}v^{2}+{\dot {m}}gh+{\dot {H}})_{i,in}-\Sigma ({\frac {1}{2}}{\dot {m}}v^{2}+{\dot {m}}gh+{\dot {H}})_{i,out}+\Sigma {\dot {Q}}_{j}+{\dot {W}}_{s}=0}$
Some important points:
1. If the system is closed AND at steady state that means the total heat flow must equal the total work done in magnitude, and be opposite in sign. However, according to another law of thermodynamics, the second law, it is impossible to change ALL of the heat flow into work, even in the most ideal case.
2. In an adiabatic system with no work done, the total amount of energy carried by mass flows is equal between those flowing in and those flowing out. However, that DOES NOT imply that the temperature remains the same, as we will see in a later section. Some substances have a greater capacity to hold heat than others, hence the term heat capacity.
3. If the conditions inside the system change over time, then we CANNOT use this form of the energy balance. The next section has information on what to do in the case that the energetics of the system change.
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2017-01-24 15:09:13
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https://www.gamedev.net/forums/topic/66447-header-file-order/
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#### Archived
This topic is now archived and is closed to further replies.
# Header file order
## Recommended Posts
Hello, I was just wondering whether or not including all your header files into one header and then including that one header in all your files is efficient or not. Does it increase your executable size? Bloat the code? Or is it more efficient than having multiple #include's in one *.cpp file. Here is my example (it's in the stdafx.h header):
#define WIN32_LEAN_AND_MEAN // Exclude rarely-used stuff from Windows headers // Include necessary header files #include #include // Header file for the OpenGL32 library #include // Header file for the Glu32 library #include // Holds the file manipulation routines #include // Holds other routines (i.e. rand()) #include "GLTimer.h" // Holds a frames per second timer class #include "GLWindow.h" // Holds the information about the window #include "GLImage.h" // Contains the GLImage class #include "GLMain.h" // Holds the main entry point of the OpenGL application #include "GLLog.h" // Holds a class that outputs log files #include "resource.h" // Holds the resource ID's of various items extern GLLog eLog; // Prevents compiler errors - defined later (in stdafx.cpp)
This makes it so each other header files doesn't need to have a #include in it, and any global variables (I try not to use them) I include in stdafx.cpp like the eLog. All my header files have:
#ifndef CLASS_NAME #define CLASS_NAME class definition without any #include's ... ... #endif
So, all I have to do in a .cpp file is:
// Include neccessary header files #include "stdafx.h" // Header file for Windows .. do the functions etc.
Is this an efficient way of linking your code together? Or should I just #include the necessary headers in the *.cpp files. Any suggestions would be greatly appreciated, EDIT: Source thing is acting strange... EDIT: Really strange... EDIT: I think I fixed it, the comments aren't aligned though EDIT: Another one... EDIT: Sorry about all the edits, the format just wasn't right Edited by - Viscous-Flow on November 9, 2001 4:31:43 PM Edited by - Viscous-Flow on November 9, 2001 4:32:51 PM Edited by - Viscous-Flow on November 9, 2001 4:34:22 PM Edited by - Viscous-Flow on November 9, 2001 4:35:18 PM Edited by - Viscous-Flow on November 9, 2001 4:36:13 PM Edited by - Viscous-Flow on November 9, 2001 4:36:56 PM Edited by - Viscous-Flow on November 9, 2001 4:37:59 PM
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The problem with adding non "standard library" includes (essentially your own headers) to your PCH header is that whenever you make a change to your own source or header files, those portions of the PCH have to be rebuilt (essentially negating the gain). Don''t put files in a PCH header just so you don''t have to include them everywhere else; put them there because they are extremely unlikely to change.
Also, including headers in a file that doesn''t need them lengthens the compilation time, because it takes longer to fully preprocess the file. The preprocessor actually substitutes the entire text of an included file in place of the #include directive before handing the file over to the compiler. In other words, unnecessary includes make the file larger and force the compiler to deal with more declarations and/or definitions that are unnecessary for the compilation of that file.
In conclusion, only put includes in the PCH header file that are required by *every* (or nearly every) source file in the project.
I wanna work for Microsoft!
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Ok, thanks Oluseyi.
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2017-11-23 03:59:12
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http://fourthfridays.org/Left%20Riemann%20Sum
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Left Riemann Sum :: fourthfridays.org
24/12/2019 · The Riemann Sum formula provides a precise definition of the definite integral as the limit of an infinite series. The Riemann Sum formula is as follows: Below are the steps for approximating an integral using six rectangles: Increase the number of rectangles n to create a better approximation: Simplify this formula by factoring. Loading. Left Riemann Sum.
If you're behind a web filter, please make sure that the domains. and. are unblocked. The calculator will approximate the definite integral using the Riemann sum and sample points of your choice: left endpoints, right endpoints, midpoints, and trapezoids. Show Instructions In general, you can skip the multiplication sign, so `5x` is equivalent to `5x`.
How to find the left riemann sum using a for loop?. Learn more about left sum, left riemann sum, for loop, riemann sum. By comparing the sum we wrote for Forward Euler equation 8 from the Forward Euler page and the left Riemann sum \eqrefleft_riemann, we should be able to convince ourselves that they are the same when the initial condition is zero. 21/01/2018 · BEWARE This TALK Will Make You RETHINK YOUR ENTIRE LIFE AND WORK life changer - Duration: 16:42. Inspire Discipline Recommended for you.
is called a Riemann sum for a given function and partition, and the value is called the mesh size of the partition. If the limit of the Riemann sums exists as, this limit is known as the Riemann integral of over the interval. The shaded areas in the above plots show. A Riemann sum is an approximation of the area under a mathematical curve between two X values. This area is approximated using a series of rectangles that have a width of delta X, which is chosen, and a height that is derived from the function in question, fX. The smaller delta X is, the more accurate the.
Riemann Sum Calculator. Topic: Area, Upper and Lower Sum or Riemann Sum. How do you calculate the left and right Riemann sum for the given function over the interval [1,5], using n=4 forfx= 3x?
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2021-04-20 01:33:09
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https://math.stackexchange.com/questions/261145/find-an-abelian-infinite-group-such-that-every-proper-subgroup-is-finite
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# Find an abelian infinite group such that every proper subgroup is finite
I found this question in Arhangel'skii and Tkachenko's book Topological Groups and Related Structures. The first chapter of the book is devoted to algebraic preliminaries.
The question actually reads:
Give an example of an infinite abelian group all proper subgroups of which are finite.
What I have done is: Every element of this group has finite order, else we could find an infinite proper subgroup, namely the group generated by $x²$ if $x$ has infinite order.
I think this can be strengthened: every element should have a prime order. Although I haven't proved this.
Intuitively this group cannot be and infinite product of smaller groups, because you could take the product of the even group factors and find an infinite proper subgroup.
Well, this is it, a highly non-trivial problem. Thanks in advance.
• You could look up the construction of a Tarski monster. – Brett Frankel Dec 18 '12 at 2:50
• wow that answers it! Do you know where I can find such a construction? – Henrique Tyrrell Dec 18 '12 at 2:53
• "Every element of this group has finite order, else we could find an infinite proper subgroup, namely the group generated by $x^2$ if $x$ has infinite order." How can you be certain that $<x^2>$ is a proper subgroup? Could it be the original group itself? – Code-Guru Dec 18 '12 at 3:20
• Tarski Monsters are not abelian groups, so they are not relevant to this problem. – Derek Holt Dec 18 '12 at 8:30
• They may not be abelian, but they have the most desired property. So it's a very good partial answer. – Henrique Tyrrell Dec 18 '12 at 12:49
More generally, you can show that the abelian groups whose proper subgroups are finite are precisely the Prüfer groups $\mathbb{Z}[p^{\infty}]$.
(Mentioned in Kaplansky's book, Infinite abelian groups, exercice 23.)
• now is this a full answer? – Henrique Tyrrell Oct 22 '13 at 17:07
Consider the set of all $2^n$-th roots of unity, as $n$ ranges over the non-negative integers. An infinite subgroup involves elements of arbitrarily high order, which generate everything below them.
• Just checked. This is a perfect answer. So the problem was trivial after all. – Henrique Tyrrell Dec 18 '12 at 17:43
• In retrospect, at least not hard, if one thinks about the kind of tower we need. – André Nicolas Dec 18 '12 at 17:48
• Did not go that deep in group theory. just starting the incantations on that realm. – Henrique Tyrrell Dec 18 '12 at 20:47
• Now, does anyone know where I can find the construction of them Tarski monsters? – Henrique Tyrrell Dec 18 '12 at 20:47
• @leo: There are $m$ $m$-th roots of unity, so for any infinite subgroup $A$, there must be a sequence $2{n_1}\lt 2^{n_2}\lt \cdots$ such that $A$ contains a primitive $2^{n_i}$-th root of unity. But if $k\lt n_i$, and $g$ is a primitive $2^{n_i}$-th root of unity, and $a$ is a $2^k$-th root of unity, then $a$ is a power of $g$. – André Nicolas May 2 '13 at 21:29
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2020-04-01 09:26:06
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https://itprospt.com/num/19630391/a-displacement-vector-d-is-given-as-40-0-m-at-an-angle-of
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5
# A displacement vector D is given as 40.0 m at an angle of 60.0 degrees EAST of NORTH: The Dx component and the component of the vector are...
## Question
###### A displacement vector D is given as 40.0 m at an angle of 60.0 degrees EAST of NORTH: The Dx component and the component of the vector are
A displacement vector D is given as 40.0 m at an angle of 60.0 degrees EAST of NORTH: The Dx component and the component of the vector are
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2022-10-02 20:06:37
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https://en.wikipedia.org/wiki/Talk:Vector_space/Archive_3
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# Talk:Vector space/Archive 3
Archive 2 Archive 3 Archive 4
## Riemann integral and completeness
In the section "Banach spaces" it is written "If one uses the Riemann integral instead, the space is not complete, which may be seen as a justification for Lebesgue's integration theory. Indeed, the Dirichlet function of the rationals, is the limit of Riemann-integrable functions, but not itself Riemann-integrable.[57]"
The first phrase if OK with me, but the second is not. The Dirichlet function is not an example of an element of the complete Banach space Lp not belonging to the (linear, non-closed) subspace of Riemann integrable functions. Indeed, the Dirichlet function is equivalent to a constant. Any sequence of Riemann integrable functions that converges to the Dirichlet function IN NORM, converges to a constant in norm. Pointwise convergence is not relevant here. Boris Tsirelson (talk) 07:45, 19 November 2008 (UTC)
In addition: "One example of such a function is the indicator function of the rational numbers, also known as the Dirichlet function." (quote from Dirichlet function) Either say "Dirichlet function", or "indicator function of the rationals", but not "Dirichlet function of the rationals". Boris Tsirelson (talk) 07:50, 19 November 2008 (UTC)
Thanks, Boris. I'm just a stupid guy :( -- I forgot that identification business at that point. Actually the problem is, I did not find a reference for the fact that the Riemann integral yields an incomplete space. It sounds like you might have one? Could you please help out by putting a precise ref at that place? For the moment I simply removed the wrong statement, which also solves your secound point. Jakob.scholbach (talk) 08:58, 19 November 2008 (UTC)
A function is Riemann integrable (on a finite interval, I mean) if and only if it is bounded, and continuous almost everywhere. The space Lp evidently contains unbounded functions (unless p=infinity), which makes the statement trivial. However, this is a very cheap argument; usually for an unbounded function one uses improper Riemann integral. It is much more interesting to see a function that fails to be continuous almost everywhere, and cannot be "repaired" by a change on a null set. The indicator of a dense open set of small measure fits. I'll try to find an appropriate reference. Boris Tsirelson (talk) 19:23, 19 November 2008 (UTC)
See Smith-Volterra-Cantor set; its indicator function fits (and its complement is a dense open set not of full measure).
Also, look at this quote: "Many functions in L2 of Lebesgue measure, being unbounded, cannot be integrated with the classical Riemann integral. So spaces of Riemann integrable functions would not be complete in the L2 norm, and the orthogonal decomposition would not apply to them. This shows one of the advantages of Lebesgue integration." Richard M. Dudley, "Real analysis and probability", 1989 (see Sect. 5.3, page 125).
For now I do not have anything better; maybe tomorrow... Boris Tsirelson (talk) 20:02, 19 November 2008 (UTC)
## Distributions
"Distributions" section starts with "A distribution (or generalized function) is a map assigning a number to functions in a given vector space, in a continuous way". First of all the map is linear (and second, continuous). Also, to be continuous (or not), it needs to be defined on a space with a topology, not just a vector space. Also, a continuous linear functional on SOME linear topological (or Hilbert, etc) space is not at all a distribution (or generalized function). Boris Tsirelson (talk) 08:05, 19 November 2008 (UTC)
That should be OK now? As you see, I'm not at all into analysis, so I'd be grateful if you could give the article a thorough scan (in these respects or any other, too). Jakob.scholbach (talk) 09:51, 19 November 2008 (UTC)
Yes, it is OK with me now. Yes, I did scan (maybe, not quite thoroughly). Really, I like the article. Boris Tsirelson (talk) 19:25, 19 November 2008 (UTC)
## Complex and real vector spaces
Maybe it would be nice to add the following (easy) example illustrating how the dimension of a space depends also on the field over which the vector space is defined:
The complex numbers over the complex field and R2 over the field of real numbers have dimensions 1 and 2 respectively.
I can add this example tomorrow (but not know; its past my bedtime).
Topology Expert (talk) 13:18, 21 November 2008 (UTC)
I have now written a word about C over R. Jakob.scholbach (talk) 21:44, 26 November 2008 (UTC)
## Topological aspects of the article
The article is great but there are some problems with the topological part of the article. For instance, 'more generally, the Grassmannian manifold consists of linear subspaces of higher (fixed) dimension n' is mathematically incorrect. In general, the collection of all such subspaces need not be a manifold (Banach manifold perhaps if restrictions on the vector space are imposed but not a Euclidean manifold). I have added a bit of information on tangent bundles but a little more could be added.
Also, if the article discusses applications of vector spaces to topology, why not include something on Banach manifolds? They are very important (in my opinion) and since they are related to 'Grassmannians for arbitrary Banach spaces', it maybe useful to include something about it.
Topology Expert (talk) 18:40, 3 December 2008 (UTC)
I have added "finite-dimensional" to the projective space discussion (which also sets the stage for the Grassmannian). As for your other additions: I think the discussion of parallelizable leeds us astray, so I have trimmed it down a bit. (The material would be an addition to tangent bundle or tangent space, but we have to stay very focussed here). Banach manifolds? Hm, currently we don't really talk about "usual" manifolds (and I don't yet see why we should). What particular application do you have in mind? Jakob.scholbach (talk) 19:00, 3 December 2008 (UTC)
## Category of vector spaces
Perhaps a brief summary of this?
Topology Expert (talk) 19:23, 3 December 2008 (UTC)
Well, what particularly do you mean? The category of vector spaces is mentioned. Of the particular properties, we could mention semisimplicity perhaps. Jakob.scholbach (talk) 21:04, 3 December 2008 (UTC)
Maybe not; I did not know that there was an article on that. I will add that to the 'see also' section.
Topology Expert (talk) 07:31, 4 December 2008 (UTC)
## Minor details
The article is coming along but there are still a few minor facts that should be added here and there. For instance, there was no mention of what an eigenspace is or what it means for a map between vector spaces to be 'orientation preserving'. I will try to add as much as I can (minor facts) but since I can't spot all minor facts it would be helpful if other editors helped. As I mentioned, the 'topology sections' could be improved but I can do that.
Topology Expert (talk) 09:28, 4 December 2008 (UTC)
## Should be a good article
In my opinion, the article should be a good article (I don't understand why it is not a featured article but I can take User:Jakob.scholbach's word on that). It has over a 100 references (for even the trivial statements) and basically anything I can think of related to vector spaces is included in the article (in all branches of mathematics). Maybe there are a few minor details that the article is missing out on but those would be probably required at the featured article nomination.
Topology Expert (talk) 09:45, 4 December 2008 (UTC)
## Manifolds and tangent spaces
The section on manifolds contains the following sentence:
"It (the tangent space) can be calculated by the partial derivatives of equations defining M inside the ambiant space."
There are many things wrong with this sentences (besides the misspelling of ambient). First of all it suggest that all manifolds have some ambient space in which they are embedded. This a popular intuitive misconception that we definitely don't want advertise on wikipedia. (This misconception is a great obstruction to people understanding the curving of spacetime in general relativity.) Of course, defining the tangent space for an intrinsic way is notoriously abstract, and I see that we don't want to talk about germ_(mathematics) in this article. But even if you accept an embedding space for the manifold this sentence makes very little sense. You can either take partial derivatives of the embedding function to find the tangent space. (although that seems awkward in this context because for an embedding you'd first need to define an abstract manifold. Or you can linearize the equations defining the manifold (i.e. x2 + y2 =1 for a circle) around a point on the manifold to find the tangent space at that point. The later clearly involves partial derivatives, but I certainly wouldn't describe it as calculating by the partial derivatives of the equations. I'm not sure how to fix this in an easy war, since most of the solutions involve going into detail on a subject that is increasingly off-topic for this subject. Does anybody see an elegant concise way to rephrase this, so that it remains understandable for most people. (TimothyRias (talk) 10:33, 4 December 2008 (UTC))
Yeah. We definitely don't want to talk about non-embedded manifolds! This is just to give some idea what this is good for. But we could just remove that sentence. Jakob.scholbach (talk) 13:05, 4 December 2008 (UTC)
I agree with that error (I hate restricitive mathematics which only considers subspaces of Euclidean space but I am sure that in the next century maths will not be like that anymore). The topology part of the article does need some work before it can go to GA (everything else is fine except for the occassional mistake such as allowing the zero vector to be an eigenvector which someone corrected recently).
Topology Expert (talk) 08:16, 5 December 2008 (UTC)
By the way, it should be mentioned that differential topology is not about "calculating" partial derivatives; it is more about checking relations between differential manifolds (as Cantor once said: mathematics is not about studying objects but rather the relations between them).
Topology Expert (talk) 08:18, 5 December 2008 (UTC)
## Recent edits
As the collaborative aspects of WP gains speed, which is cool, I take the opportunity to point out some ideas I have about writing a good article, by exemplifying with some recent edits. My ideas have been shaped by/after FAC discussions, manual of style and so on. I don't want to be imposing, but am just trying to save time for all of us.
• Typesetting is something which requires care, e.g. '''R<sup>2</sup>''' (R2) should be '''R''<sup>2</sup> (R2).
• Italics are used only for top-level notions, or to emphasize things: "The determinant of a square matrix can also tell us whether the corresponding linear transformation is orientation preserving or not." I feel orientation preserving is neither of the two.
• Talking about "us" and "we" should be avoided.
• Please try to keep the structure of sections etc., if you agree with it. E.g. adding another example (cross product to the Lie algebra thread) should be close to the examples already given. If you want to reorganize things (in this case put examples up first), look what other changes this necessitates (in this case moving the other example up)
• Whereever {{Main}} templates are used, the corresponding subarticle should not be wikilinked again, to avoid overlinking. Also, main templates should link to only very important related articles (which cross product is not, IMO).
• The "see also" section should not repeat items already covered in the text. Jakob.scholbach (talk) 13:09, 4 December 2008 (UTC)
Sorry if I was not following some of those conventions; I will try so in future. But if I ever miss a convention, feel free to either correct it or revert (I will try my best not to miss one) (I know it is hard to follow my edits like that but hopefully 90% of them, at least, should be alright).
Topology Expert (talk) 08:12, 5 December 2008 (UTC)
Thanks for cleaning it up.
Topology Expert (talk) 13:10, 5 December 2008 (UTC)
Personally, I do not agree with this convention for several reasons:
1. A reader may not read a particular section where a topic is Wikilinked. Often only the first occurrence of a topic is Wikilinked, so a reader of a later section will be unaware that there exists a Wikilink.
2. I personally find it very handy to be able to scroll down to See also just to see what is out there. If significant subjects are not there, it's a problem.
3. As an editor, when changing items in an article I often wish to refer to other related articles to be sure of compatibility and not missing items of importance. It is nice to use See also for this purpose, rather than scanning through a long article to find all the embedded links.
For all these reasons, I believe all significant articles should appear in See also section or referred using a {{seealso}} template. Brews ohare (talk) 15:54, 6 December 2008 (UTC)
I disagree with you. We have to distinguish between an article which is under development, i.e. a stub or start class article, and an article that is reasonably complete, such as this one here. When writing a stub, it is very good to put related notions into the s.a. section, just as a replacement of a more thorough treatment.
If interpreted literally, your arguments (all of which are pretty much parallel) would lead to including pretty much every linked notion in the s.a. section, which is not useful either for editors nor readers. It is, sad or not, a fact that a reader will have to read an article if she/he wants to know about it. If you have little time, a glance at the TOC should give you the big points of the topic in question. See also ;) the relevant MOS section. The s.a. section is just for putting peripherical notions whose (minor) importance (to the article in question) does not give them the "right" to deserve a subsection or even a phrase. Jakob.scholbach (talk) 16:07, 6 December 2008 (UTC)
Jakob exaggerated my suggestion, which actually states:
I believe all significant articles should appear in See also section or referred using a {{seealso}} template.
That means I'd object to putting "an obscure link to a section" in See also, but favor including "significant articles", unless already in a {{seealso}} or {{main}} template. Of course, who can argue against using judgment? Brews ohare (talk) 17:55, 6 December 2008 (UTC)
I'm sorry, Brews. Somehow I did indeed not see your last line above. Do we agree that category of vector spaces (just as an example) should not reappear in the see also section, or do you think it is significant enough to make it show up again? I guess it's also not that important of an issue. Much more enerving (to me) is it that despite my iterated posting at WT:WPM nobody seems to be inclined to review the article. What can we do about that? Jakob.scholbach (talk) 18:14, 6 December 2008 (UTC)
Hi Jacob: I have no experience with such things. Try asking User_talk:Dicklyon, who I have found to be very helpful. Brews ohare (talk) 20:23, 6 December 2008 (UTC)
It can be a useful discipline in the See also section to use headings to classify various links by subject. Doing this helps the reader and also leads to some useful scrutiny of what is linked, to avoid it becoming a "gee, this is interesting" section. Here is an example from k·p perturbation theory:
Multiple column format is implemented using {{Col-begin}} rather than a myriad of alternatives because many of the alternatives work fine in Firefox, but not in Internet Explorer. Brews ohare (talk) 15:59, 6 December 2008 (UTC)
## notes from GeometryGirl (talk)
• I don't really know how to deal with this sentence: "Another conceptually important point is that elements of vector spaces are not usually expressed as linear combinations of a particular set of vectors". "point" sounds informal and something is wrong with "usually expressed"
• In the section "linear equations" I would add a small note about annihilators being 'dual' to linear equations
• What do you mean by that? Jakob.scholbach (talk) 12:55, 7 December 2008 (UTC)
• Well, taking the example given, if e1, e2, e3 is the standard basis for (R^3)* (the dual of R^3) then the space of solutions is simply the annihilator of W = <e1 + 3e2 + e3, 4e1 + 2e2 + 2e3>. The dual perspective makes it clear why the set of solutions to a set of linear equations is naturally a vector space. GeometryGirl (talk) 14:50, 7 December 2008 (UTC)
• Hm. I'm not so convinced. That the solutions form a vector space is clear(er) by seeing it as the kernel, right? I personally believe talking about annihilators would lead us a bit astray. We would have to talk about dual space, dual basis, pairing at that point, which is a bit too much. What do others think? Jakob.scholbach (talk) 15:54, 7 December 2008 (UTC)
## GA Review
GA review (see here for criteria)
1. It is reasonably well written.
a (prose): b (MoS):
2. It is factually accurate and verifiable.
a (references): b (citations to reliable sources): c (OR):
3. It is broad in its coverage.
a (major aspects): b (focused):
4. It follows the neutral point of view policy.
a (fair representation): b (all significant views):
5. It is stable.
6. It contains images, where possible, to illustrate the topic.
a (tagged and captioned): b lack of images (does not in itself exclude GA): c (non-free images have fair use rationales):
7. Overall:
a Pass/Fail:
• You pass. Congratulations! Ozob (talk) 03:33, 12 December 2008 (UTC)
Here are some specific issues that I'd like fixed before this reaches GA:
• The lead says "much of their [vector spaces'] theory is of a linear nature"; but I don't think the meaning of "linear nature" will be apparent to people unfamiliar with vector spaces. E.g., someone might not know what a linear combination or linear transformation is.
• OK. Better now?
• Yes.
• In the "Motivation and definition" and definition section, "linear combination" has not yet been defined. It might be better to say that there is no preferred set of numbers for a vector, and to say no more until bases have been introduced.
• OK.
• Also good.
• In the subheading "Field extensions", the description of Q(z) is odd: It sounds like you mean for z to be a transcendental, but you say that z is complex. If z=1 then the field extension is trivial; even if the field extension is non-trivial, it's not unique (square root of 2 vs. cube root of 2). I see below that you do really mean for z to be complex, but perhaps there's a better way to say what you mean.
• I'm not sure I understand your points. I do mean z to be complex, just for concreteness. ("Another example is Q(z), the smallest field containing the rationals and some complex number z.") What is the problem with z=1 and a trivial extension? What do you mean by "it's not unique"? (I think, for simplicity, the subfield-of-C-definition I'm giving is appropriate at this stage, and yields something unique).
• I think what bothers me is that you say you are about to give another example (singular) and then proceed to give a family of examples (plural). I've changed the text to try to make this better; is this OK for you? (BTW, I used an α instead of a z because z looks like a transcendental to me. This might have been part of my confusion, too. But change it back if you think having a z is better.)
• α is fine, but I will have to eliminate the redundancy with the section on dimension later. Jakob.scholbach (talk) 08:07, 10 December 2008 (UTC)
• The article should say very early that abstract vector spaces don't have a notion of an angle or of distance or of nearness. This is confusing for most people.
• OK. (In the definition section).
• The bolded expression〈x | y〉does not display properly on Safari 3.0.4; the left and right hand angle brackets show up as squares, Safari's usual notation for "I don't have this character". (It works when unbolded, as I found out when I previewed this page.)
• Yeah, it was weird, there were two types of angle brackets. Can you read them now (there are three occurences in that section).
• Yes.
• The natural map V → V** is only discussed in the topological setting. It should be discussed in general. (Note that the map is always injective if one considers the algebraic dual (for each v, use the linear functional "project onto v".))
• Done. (will provide a ref. later) Jakob.scholbach (talk) 12:50, 7 December 2008 (UTC)
• I now had a look at most of the algebra books listed in the article and none of them, actually, talks about algebraic biduals. So I wonder if this is so important. (I wondered already before). Jakob.scholbach (talk) 21:40, 9 December 2008 (UTC)
• Hmm! I know that they appear in Halmos's Finite dimensional vector spaces (p. 28, exercise 9). But it seems to me that the best reason for discussing them is the finite-dimensional case: Right now, the article doesn't discuss reflexivity of finite-dimensional vector spaces, a real gap!
• JPEG uses a discrete cosine transform, not a discrete Fourier transform.
• OK. Jakob.scholbach (talk) 12:04, 7 December 2008 (UTC)
• I'm almost ready to say this is a GA; my only outstanding issue is reflexivity of finite dimensional vector spaces. Ozob (talk) 00:36, 10 December 2008 (UTC)
• What else do you want ("... This [i.e. reflexifity for top. v.sp.] is in contrast to the linear-algebraic bidual, i.e. where no continuity requirements are imposed:... ")? Jakob.scholbach (talk) 15:53, 10 December 2008 (UTC)
• Well, I'm not sure what's the best way to state it. But I feel like the fact that all finite-dimensional spaces are reflexive is really, really important and needs to be mentioned somewhere. The way you have it now is fine. Ozob (talk) 03:36, 12 December 2008 (UTC)
Here are some other issues which aren't as pressing but which I think you should handle before FA:
• I'm not sure that likening a basis for a vector space to generators for a group or a basis for a topology will help most readers. Most people who use and need linear algebra have never heard of these.
• I'm not sure either! I removed it.
• Since you mention the determinant, it's worth mentioning that it's a construction from multilinear algebra. A sentence or two should suffice.
• Except for det (f: V → V) being related to Λ f: Λ V → Λ V (which I think should not be touched here), I don't see why the determinant belongs to multilinear algebra. What specifically do you think of?
• That's exactly what I was thinking of. I don't want to make a big deal about that construction, but I do think it's good to mention—it's the right way to think about the determinant, and the only way I can think of which admits generalizations (e.g. to vector bundles). I put a sentence in the article about this.
• It seems that for most of the article, whenever you need an example of a non-abstract vector space, you use solutions to differential equations. I agree wholeheartedly that these are important, but there are probably other good examples out there which shouldn't be slighted.
• Hm, I also talk about (non-differential) equations, but what else do you have in mind?
• I'm not sure! I was hoping someone else here would have good ideas.
• It also seems that you rely on convergence to justify the introduction of other structures such inner products; but inner products can be (and should be, I think) justified on geometric terms, because they're necessary to define the notion of an angle.
• OK, you did this.
• It's also worth mentioning the use of vector spaces in representation theory.
• Done. (Very briefly).
• Good, that's as much on representation theory as this article needs. Ozob (talk) 00:47, 10 December 2008 (UTC)
• When writing an integral such as ${\displaystyle \int f(x)dx}$, the output looks better if you put a thinspace (a \,) between f and dx: ${\displaystyle \int f(x)\,dx}$.
• OK.
• Image:Moebiusstrip.png should be an SVG.
• I tried to convert the image into an svg (Image:Moebiusstrip.svg), but somehow the strip (which was taken from a photo, so png previously) is invisible to me?! Any ideas about that? Jakob.scholbach (talk) 12:50, 7 December 2008 (UTC)
I'll try to make a SVG picture of a moebius strip later today. (TimothyRias (talk) 10:58, 8 December 2008 (UTC))
Ozob (talk) 02:40, 7 December 2008 (UTC)
Thanks very much, Ozob, for your review! Jakob.scholbach (talk) 12:04, 7 December 2008 (UTC)
I concur with the comments above; I have the following comment to make on tensor products, which I would like to be taken addressed before GA status:
The description of tensor product as it stands is too vague (such as "mimicking bilinearity"). It would be better to first give the universal property of tensor product of V and W as the unique vector space E + bilinear map V × WE with the universal property of expressing all bilinear maps from V × W to a vector space F as a linear maps from E to F. Then one could state that a space with these properties does exits, and outline the construction. Similarly the adjoint property of tensor product with respect to Hom is too vague. To control article size, one could consider leaving that out as tensor product article is wikilinked; otherwise one should definitely point out that tensor product is a (bi-) functor. As for extension (and restriction) of scalars (tensoring with extension field of the base field), that could be treated, but then again functoriality of tensor product would be natural to include. Perhaps effective use of summary style could help keep amount of material here still manageable. Stca74 (talk) 10:45, 7 December 2008 (UTC)
Thank you too, Stca, for your review: I have trimmed down the tensor product discussion a bit, but also made it more concrete. I think doing the universal property thing properly (i.e. with explanation) is too long and also a bit too complicated (even uninteresting?) for general folks, so should be deferred to the subpage. As for the isomorphism: I don't know why I called this adjunction isomorphism, since it is effectively both adjunction and reflexivity of f.d. spaces. Anyhow, this comment was just to put tensors in line with scalars, vectors and matrices, but I would not go into functoriality etc. Jakob.scholbach (talk) 15:12, 7 December 2008 (UTC)
Looks more precise now. However, I would still consider adding the universal property (perhaps somewhat informally, at least) - as least for me it is the only way to make sense of the construction, which otherwise risks being just a tangle of formulas. As for the last few lines after the representation of Hom as tensor product of dual of the domain with the target, I'm not sure if I can follow (or expect others to follow). Actually, the canonical map goes in general from the tensor product into the Hom space and is injective. It is bijective if one of the spaces is finite dimensional. Thus, if you insist, you get an interpretation of a tensor (element of the tensor product) as a matrix, but not really tensor as a generalisation of matrix (following scalar, vector, matrix list). Stca74 (talk) 20:06, 7 December 2008 (UTC)
OK, I scrapped the sketched ladder of "tensority". Also the universal property should be fine now. Jakob.scholbach (talk) 21:40, 9 December 2008 (UTC)
Unfortunately, the tensor product section now has a problem: It doesn't define "bilinear", so it doesn't make a lot of sense. The previous version was better in this respect because it was only hand-waving, so the reader didn't expect to understand; but now that the article is more precise, the lack of definition of "bilinear" is a problem. I'm not really sure what to do here; if one defines "bilinear" then one should give an example, but the simplest example is the dot product, which is later in the article. And being vague, as Stca74 noted, is no solution either. It might be good to introduce the dot product here and then reintroduce it later in the inner product section; the second time you'd point out that it's positive definite. (Also, the inner product section currently calls the Minkowski form an "inner product" even though it's not positive definite. I know that in physics, "inner product" doesn't mean positive definite, but it certainly does in math. This deserves a remark somewhere, I think.) Ozob (talk) 00:55, 10 December 2008 (UTC)
(<-) Well, the bilinearity is certainly no problem. I mentioned this now. The problem is more: how to create a little subsection that is inviting enough to guide the reader to subarticle. When I learnt this, I kept wondering "what is this u.pr. all about?" I only got it after learning about fiber products of (affine) schemes, but we certainly cannot put that up! Jakob.scholbach (talk) 08:07, 10 December 2008 (UTC)
Oof, that's a tough way of figuring it out! (Not that I did better!) I agree, this is a tough thing to work out. It'll have to be done before FA, though (if that's where you want to take the article). The only really elementary context I can think of where they turn up is bilinear forms. It might be best to have a section on bilinear forms first (which would mention inner products and the Minkowski metric and link to the article on signature) and then use those to justify tensor products: "Tensor products let us talk about bilinear maps, which you now know to be wonderful, in terms of linear maps, which you also know to be wonderful." That would require reorganizing the article a little, but I don't see a good other solution. Ozob (talk) 03:45, 12 December 2008 (UTC)
• The article should say very early that abstract vector spaces don't have a notion of an angle or of distance or of nearness. This is confusing for most people.
Maybe you should not emphasize this (nor should you write that they do have these structures) because you can equip vector spaces with a norm (distance and nearness) or an inner product (for angles) and I am quite sure that most of mathematics done on vector spaces studies these structures on them (such as Banach space theory or Riemannian geometry). So perhaps keeping the sentence, should imply that there is an explanation that you still can equip these structures on vector spaces since these strucutures are indeed very important in mathematics.
Topology Expert (talk) 17:17, 7 December 2008 (UTC)
Topology Expert (talk) 17:17, 7 December 2008 (UTC)
Careful with generalisations: any absolute value on a finite field is improper (|x|=1 for all non-zero x) and thus there are no interesting norms to put on vector spaces over finite fields. And while norms on finite-dimensional real vector spaces equivalent, there are still no canonical norms nor inner products. I do agree with Ozob's view that it makes sense to warn readers about this potentially counterintuitive fact. Stca74 (talk) 20:06, 7 December 2008 (UTC)
## Topological vector spaces and biduality
I'm afraid the discussion on biduals (discussed already above during GA nomination) in the topological context is a bit too inaccurate as it stands, and contains claims which only hold with additional hypotheses.
First, the definition of the bidual is incomplete unless the topology of the dual is specified - there is in general no preferred topology on the (topological) dual. The bidual E ' ' is the (topological) dual of the strong dual of E. The theory is normally developed only in the context of locally convex spaces, for which it indeed follows from Hahn-Banach that the canonical mapping of E into its bidual is injective precisely when E is Hausdorff. Next, it is possible to define semireflexive spaces as the ones for which this canonical map is bijective without specifying a topology on the bidual. Reflexivity refers to the canonical map being an isomorphism when the bidual is given the strong topology (with respect to the strong topology of the dual of E). For normed spaces semireflexive and reflexive are equivalent.
As the above suggests, the concept of duality for general (locally convex) spaces is not entirely straightforward, and it would be better to avoid introducing too much of the theory in this article which is primarily about the algebraic theory of vector spaces. While it could in principle be feasible to discuss biduals more easily for normed vector spaces, I would rather agree with Ozob's comments in the GA discussion and treat biduals and reflexivity in the purely algebraic setting here. It would then be possible to point the reader to the relevan articles on topological vector spaces for the related concept in that context. For a very clear discussion on biduals in the algebraic context (also for general modules, not only vector spaces) see e.g. Bourbaki, Algebra, Ch II. Stca74 (talk) 20:39, 12 December 2008 (UTC)
OK. First, what is written certainly has be correct and as precise, so this has to be amended. From what I know, though, I cannot see why algebraic biduality is so important or even more important than topological biduality. The algebraic statement is a triviality, whereas the topological one is not at all. Also, I think, the article should not give more weight to algebraic assets than to functional analysis etc. So, some concrete questions to everybody:
• What makes algebraic biduality so important? (I don't have the Bourbaki at hand right now, but I'm suspecting it does not tell too much about its importance).
• Is it right to think of the strong topology on the dual as the "most natural one"? Jakob.scholbach (talk) 09:44, 13 December 2008 (UTC)
I don't know enough functional analysis to answer your questions, in particular I wouldn't know what the strong topology on the dual is in the general setting. I do think though that (everything else being equal) algebraic concepts should be stressed in this article, because "vector space" is an algebraic concept. But why do we talk about biduals at all? Isn't it a bit far removed?
The paragraph does make a nice point that for topological vector spaces, you want to talk about the continuous dual instead of the algebraic dual. That's worth keeping. But I'm not so sure about biduals. In what context do you want to introduce them? If you talk about locally convex vector spaces, you'll have to add yet more definitions. As mentioned before, it's easier to talk about it in the context of Banach spaces. But still I think it's a bit too far removed to be in this article. There is no Banach space theory in here, nor should there.
What do you think about adding instead an example of a topological vector space that is not a Banach space? The easiest example I know is C. -- Jitse Niesen (talk) 16:33, 13 December 2008 (UTC)
Strong operator topology says "It (the s.o.t.) is more natural too, since it is simply the topology of pointwise convergence for an operator." I'm not experienced enough either to say whether biduals are crucial, but reflexivity seems to be somewhat important. What say the functional analysts in the house? Jakob.scholbach (talk) 16:49, 13 December 2008 (UTC)
As for the non-Banach example, we mention the noncompleteness of Riemann integrable functions. I prefer this over C since it shows the superiority of Lebesgue, whose influence is all over the place in these matters, right? Jakob.scholbach (talk) 16:51, 13 December 2008 (UTC)
(←) On the importance of the algebraic bidual: while the proofs of the statements about biduality are indeed quite trivial for vector spaces, one is dealing with a special case of a much deeper algebraic concept (and one which is important even where the proofs are easy). The precisely same concept is already non-trivial for modules over rings more general than fields. In that context the canonical map to bidual is injective for projective modules and bijective for finitely generated projective modules. Via the well-known correspondence (finitely generated) projective modules have interpretation as (finite-rank) vector bundles (rightly introduced as generalisations of vector spaces in the article) , both in differential geometry over the rings of smooth functions, and in algebraic geometry for general commutative rings. In the somewhat more general set-up of coherent sheaves biduality and reflexivity are common issues to consider in the practice of algebraic geometry. Not surprisingly, the same occurs in homological algebra, where double duals of cohomology spaces, modules, sheaves are a very common occurrence. Eventually this leads to biduality in the context of derived categories as a crucial component of Grothendieck's "six operations" formalism for the very important generalisations of Poincaré / Serre -type dualities. The theory of D-modules deserves a related mention here too. Finally, one can mention the role the canonical mapping into bidual played in the formulation of natural transformations of category theory. Hence, all considered, I do think the (algebraic) bidual deserves to be intorduced briefly here, as a simple incarnation of truly important and fruitful concept.
On "natural" topologies to define on the topological dual E ' of a topological vector space E, I suppose one could argue that for normed spaces the strong topology is the most "natural": it is just the familiar norm topology where the norm of a functional f is the supremum of the absolute values of f(x) where x ranges over the unit ball (or sphere) in E. The canonical mapping EE ' ' is then always injection. However, from another viewpoint a "natural" topology T on the dual E ' would have the property that the natural map φ: EE '* to the algebraic dual of the topological dual (defined by the duality pairing E × E ' → R) were a bijection to the topological dual of E ' equipped with the topology T (i.e., T compatible with the duality). Now (under the necessary assumption that φ is injective) all topologies between the weak topology (weakest) and the Mackey topology (strongest) satisfy that condition. However, in general the strong topology of the dual is stronger than the Mackey topology; for the strong topology to be compatible with the duality (and hence equal to the Mackey topology) is precisely the condition of E being semi-reflexive. For normed spaces reflexive is equivalent to semireflexive, which shows that there is a clear viewpoint from which (for example) the weak topology of the dual of a normed space is "more natural" than the strong topology, at least when E is not reflexive. This would be the case for example for L1-spaces. But again, all of the above digression I think mainly helps to show why the topological reflexivity is best left to articles on topological vector spaces and functional analysis. Stca74 (talk) 18:53, 13 December 2008 (UTC)
Huh! Since I cannot cite this talk page ;-( I decided to trim down the presentation somewhat and moved the algebraic biduality statement up to the algebraic dual. I left the Hahn-Banach theorem but without referring to the bidual. Jakob.scholbach (talk) 19:50, 13 December 2008 (UTC)
## Tangent space edits
this edit removed some content as per "removing inaccuracies". What exactly did you mean by that, Silly rabit? I'm inclined to revert that change (it removed references, a rough description what the tangent space is, Lie algebras vs. Lie groups). Jakob.scholbach (talk) 11:27, 3 January 2009 (UTC)
The parts of the paragraph I removed were a bit confused and overstate the important of the tangent space itself. In particular, the vector flow does not go from the tangent space to the manifold: perhaps what was meant was the exponential map? I don't know. Also, the tangent space of a Lie group is quite ordinary, and doesn't "reveal" anything special about the Lie group. It is given the structure of a Lie algebra in a natural manner, but that is something extra. Another example in the same vein: does the tangent space of a Riemannian manifold reveal something special about the manifold? No, it is the metric which does that. siℓℓy rabbit (talk) 14:42, 3 January 2009 (UTC)
I sort of agree with you, but I like to play devil's advocate. Doesn't the Lie algebra determine the Lie group? (up to the connected component of the identity?) It has been a long time since I studied Lie groups and their algebra's, but I do seem to remember something along these lines. Thenub314 (talk) 14:51, 3 January 2009 (UTC)
More or less yes. Anyway, I have added the statement about Lie groups and their Lie algebras to a more conceptually appropriate place in the article that will hopefully satisfy Jacob's objection. siℓℓy rabbit (talk) 15:06, 3 January 2009 (UTC)
OK. Probably I was indeed to sloppy. Just one point: the statement "The tangent space is the best approximation to a surface" is unclear, to me and probably more so for a reader who does not yet know about the t.sp. What exactly does "best" mean? Jakob.scholbach (talk) 22:44, 3 January 2009 (UTC)
A good point. I have added an additional link to the sentence in question to linearization, and an additional content note defining precisely what is meant by "best" in this context, together with a reference. siℓℓy rabbit (talk) 02:52, 4 January 2009 (UTC)
Good, thanks. Jakob.scholbach (talk) 19:19, 4 January 2009 (UTC)
## Minor changes
I would like to make the following minor changes to the lead sentence and paragraph.
I would like to change the lead sentence to
A vector space is a mathematical structure formed by a collection of objects, called vectors, that may be added, subtracted, and scaled.
This may not be necessary. I hadn't noticed it was put in the first section until now, I kind of like it better in the lead, but it would not make me unhappy if it stays the way it is. Sorry I should read more carefully before I write. Thenub314 (talk) 08:53, 8 January 2009 (UTC)
Also in second sentence I think "Euclidean vectors" was better as just "vectors", because we follow it shortly after with the phrase "Euclidean space," and it seems like one too many Euclideans for this sentence. I suggest we change Euclidean vectors with vectorss (this is still linked to the same article), and we link plane with an appropriate article to make clear we mean the Euclidean plane. (Prehaps Plane (geometry)?)
Does anybody have a good idea what image could illustrate the vector space idea? The current image is pretty crappy, I think, for it conveys basically nothing. Jakob.scholbach (talk) 18:12, 11 January 2009 (UTC)
Added a better image with better description. Please have a look. PST
Well, I think that one is only little better than the previous one. BTW, please sign your posts at talk pages! Jakob.scholbach (talk) 19:28, 12 January 2009 (UTC)
Also, the new caption is not great, since in many vector spaces there is no, or at least no preferred inner product, therefore "closer to" some vector is meaningless. Jakob.scholbach (talk) 19:37, 12 January 2009 (UTC)
Yes, I just intended that to be a rough idea (I kind of felt uncormfortable when writing that caption). Hopefully someone will get a better image soon (the current images at commons are not very good so someone will probably have to upload one). I am quite happy to improve this article in the near future (but perhaps I should discuss here before I edit because I want to make sure that my edits are appropriate). --Point-set topologist (talk) 20:37, 12 January 2009 (UTC)
How about a drawing of the parallelogram law for adding and subtracting vectors? That's the cover illustration for Sheldon Axler's Linear Algebra Done Right. --Uncia (talk) 16:01, 15 January 2009 (UTC)
That's an idea. I'll try merging this illustration with a flag (0-, 1-, and 2-diml subspace of R^3) tonight, unless somebody else is up to it... Jakob.scholbach (talk) 16:34, 15 January 2009 (UTC)
I like the picture. There are a couple of points about the caption that I thought were not clear: (1) the gray square is not actually the vector space, because the vector space extends to infinity in all directions; (2) The label 0 is used but not explained; maybe we could add "the zero vector 0 belongs to all subspaces". --Uncia (talk) 22:45, 15 January 2009 (UTC)
Although the image is much better than before, I am not perfectly satisfied. It has one error (mentioned above) not to mention that it looks a bit messy (and hard to follow). But I think that the image is temporarily good enough. PST 09:16, 16 January 2009 (UTC)
## This is certainly the second best article, I have seen, in Wikipedia
If this goes for FA, I would be quite pleased to support. However, I am a little worried regarding the issue on 'range of content'. Vector spaces have so many applications everywhere (heaps in mathematics) and I don't think that the current article describes all of these applications. This maybe because it is not supposed to, but if it is, this is just a suggestion. More importantly, the sections regarding topology have to be cleaned up. I see that we should not go off the topic so that has to be done carefully (but note that the section on tangent bundles here is, in my view, better than the article :)). PST (--Point-set topologist (talk) 09:57, 15 January 2009 (UTC))
## Dimension
Jacob, are you seriously saying that all infinite dimensional vector spaces are isomorphic to each other? How about the Hilbert spaces? Is H0=H1 ? −Woodstone (talk) 22:57, 18 January 2009 (UTC)
I'm seriously saying that two vector spaces of the same dimension are isomorphic as vector spaces. There may be v.sp. that are both infinite-dimensional, but the cardinality of the two bases is different. Also, L^p is isomorphic to L^q as vector spaces, but not as topological vector spaces. Likewise with any other counterexample you may think of. Just see the relevant article section and the refs cited therein. Jakob.scholbach (talk) 23:00, 18 January 2009 (UTC)
That's only a half answer. Are you stating that H0 and H1 are isomorphic as vector spaces? I think not. −Woodstone (talk) 23:09, 18 January 2009 (UTC)
I don't know that notation. What does it mean? But anyway, you can answer it yourself: if the dimensions agree (as cardinal numbers) they are isomorphic as v.sp., otherwise they are not. Jakob.scholbach (talk) 23:32, 18 January 2009 (UTC)
Two vector spaces of the same dimension are isomorphic, even if that dimension is an infinite cardinal. Indeed, any vector space over a field F with a basis set X is isomorphic to ${\displaystyle \oplus _{x\in X}F}$, which is the space of all finitely supported functions XF. By precomposition with a bijection to the cardinal |X| of X, this can be put into a one-to-one linear correspondence with the vector space of finitely supported functions |X|→F. For the other question, the example of ${\displaystyle H_{0}}$ and ${\displaystyle H_{1}}$ seems strange to me, because these typically denote Sobolev spaces, in which case H0 and H1 are both separable Hilbert spaces, and so are in fact isomorphic as Hilbert spaces as well. siℓℓy rabbit (talk) 23:43, 18 January 2009 (UTC)
I have added a content note to clarify this. I am generally opposed to footnotes in the lead. However, sometimes they are necessary. This seems to be such a case. Please change the wording around and provide references as appropriate. Originally, I thought that Halmos Finite dimensional vector spaces provided some discussion of this, but I was unable to find a suitable section there. Anyway, I think the clarification would be much more effective with a suitable reference. siℓℓy rabbit (talk) 01:02, 19 January 2009 (UTC)
I think that most people are familiar with the idea of a dimension. But can any laymen who reads this discussion confirm what exactly they think it is? Tracing back to my earlier days, I used to think that higher dimensional spaces are "more complex". When writing the article, perhaps you may like to bear that in mind. --PST 08:47, 19 January 2009 (UTC)
## Reference
I added this reference sometime ago: Cohen 1998, p. 31, The Topology of Fibre Bundles, but it seems to have disappeared. Is this a problem with the formatting of the reference? I think that this PDF file has a good lot of information on vector bundles so it should be there. --PST 08:59, 19 January 2009 (UTC)
Yes, that was me. I didn't remove it primarily because of the reference, but of the statement that vector bundles form a monoid (which I thought leads a bit too far away). Remember this is an article about v.sp., not vector bundles, so we should not include any reference which is just somewhat nice on a side aspect of the article topic. Also, unless the thing is printed by a regular publisher (which it is not?), the file does not count as a reference, but as an external link, which makes it even less interesting to include here. Jakob.scholbach (talk) 16:21, 19 January 2009 (UTC)
## Inner product notation
I've seen the following notations commonly used for inner products: ${\displaystyle (u,v)}$; ${\displaystyle \langle u,v\rangle }$; ${\displaystyle \langle u|v\rangle }$ (the last only as part of Dirac notation). This article uses yet another, viz: ${\displaystyle \langle {\mathbf {u} }|{\mathbf {v} }\rangle }$, which I have never seen (but I'm a physicist). I assume this is used by some mathematicians, but how common is it? At any rate shouldn't the other notations be mentioned? PaddyLeahy (talk) 11:45, 19 January 2009 (UTC)
How exactly are the last to different? Except that the later uses both braket notation and bold to (doubly) indicate that the objects are vectors. Anyway, if different notations are to be mentioned, shouldn't at least the ordinary dot notation be mentioned? (TimothyRias (talk) 12:34, 19 January 2009 (UTC))
I don't have a preference for either of the variants. The boldface for vectors is just a general notation in this whole article (and elsewhere), that's unrelated to the inner product. The dot notation is mentioned (for the standard dot product, for which, I feel, it is preferably used). I think additional notations should not be discussed here, since this is an article about vector spaces. Jakob.scholbach (talk) 16:18, 19 January 2009 (UTC)
Still using both boldface and braket notation is weird since it doubly denotes the objects as vectors. Notation wise ${\displaystyle \langle {\mathbf {u} },{\mathbf {v} }\rangle }$ probably is nicer. (TimothyRias (talk) 22:49, 19 January 2009 (UTC))
Well, we use boldface all the time, so I don't see a reason to change it at that place. But whether we put a "|" or a comma in the middle, I don't care. Change it if you like... Jakob.scholbach (talk) 23:15, 19 January 2009 (UTC)
## When to mention fields
It seems a bit early in the lead to bring up fields, since we cover them in the definition. I have tried leaving it in, but I am concerned it might get beyond interested high school students if we jump into it too quickly. Thenub314 (talk) 15:59, 20 January 2009 (UTC)
Well, I think they are too important to be omitted, but I like the way you trimmed it (except for the use of the second person, which I removed by making that sentence passive). But I think we should at least give a non-rigorous explanation of what a field is (indeed because high school students won't know what they are[1]), such as
... provided the scalars form a field (such as rational numbers or complex numbers), that is, that they can be added and multiplied together satisfying similar properties.
, or something like that.
[1] Incredibly, I've seen junior high school books with definitions of groups, rings, and fields, but I guess more than 99.99% of all teachers simply skip that parts. -- Army1987 – Deeds, not words. 16:22, 20 January 2009 (UTC)
I did another take of the first section of the lead. I also think we should not put too much emphasis on the base field. A people who does not know about this will never think: Oh and what if I consider a complex vsp over the reals? Also, I somewhat disagree that we should explain what a field is. Again, people who don't know what a vsp is will hardly digest the brief definition "you have +, -, *, /". Indeed most of the article, and most of the applications both in mathematics and beyond concern real and complex spaces, so conveying this particular case in the lead is fairly sufficient. Jakob.scholbach (talk) 18:56, 20 January 2009 (UTC)
## Recent change to the lead sentence
I don't really like this edit. Apart from some errors (such as the inappropriate mention of mathematical physics) that could be cleared up with copyediting, I think it is actually more formal rather than more understandable for a non-mathematician. Do we have any "non-mathematicians" available for comment? Army, I believe, is an engineer or physicist. siℓℓy rabbit (talk) 02:46, 21 January 2009 (UTC)
I don't see why the mention of mathematical physics is inappropriate: that's where the motivation came from, and where virtually all the applications are. Re formality, there are basically two directions the article could go: it could start with a "physics" intuition of a vector as something with magnitude and direction, or it could start with an abstract description. If it starts with an abstract description, the previous version wasn't good enough. Saying "a vector is something that can be scaled, or multiplied by a number" is only understandable to somebody who already understands it. Looie496 (talk) 05:19, 21 January 2009 (UTC)
"Mention" of mathematical physics is appropriate, but in proportion to its prominence in the article. So far, not much of the article is dedicated to physics, and therefore the second sentence does not appear to be correct weight for this. Also, contrary to conventional dogma, the abstract notion of a vector space was not motivated directly by physics, mathematical or otherwise. (It is true that the notion of a physical vector emerged from such considerations.) siℓℓy rabbit (talk) 07:10, 21 January 2009 (UTC)
I tend to agree with siℓℓy rabbit. Though I am a mathematician, I am a pretty bad one, so hopefully my input carries some weight. I think the notion of scaling is pretty clear and intuitive for people who haven't seen vector spaces (we all have some seen scale models, drawings, etc.) The previous lead prehaps could be criticized because it inferred you could "add" some objects called "vectors". But I think the current picture next to the lead made that rather clear as well. Overall my opinion is we go back to the lead we had a day (or two) ago. Thenub314 (talk) 07:19, 21 January 2009 (UTC)
I think reverting Looie's edit there was appropriate. Do you still think the physics aspect has too much weight now? (Currently just one motivating and hopefully understandable example from physics in the lead. I plan to brush over the motivation section in this direction too, but there also highlighting the mathematical background, i.e. triples etc. of numbers). I think one motivational example in the lead is good, since then we can say that the axioms are modelled on that. Jakob.scholbach (talk) 07:30, 21 January 2009 (UTC)
I don't like the new lead either. And Looie496: "that's where the motivation came from, and where virtually all the applications are" is false. Some motivation does come from physics but there are heaps of applications of vector spaces in mathematics; perhaps as much as physics. Something about physics should be mentioned, but I strongly disagree that physics is the only reason why vector spaces were invented. --PST 07:54, 21 January 2009 (UTC)
By the way, Army is a high school math teacher as he/she noted already on the "comments" page. --PST 08:01, 21 January 2009 (UTC)
I'm not. I'm an undergraduate physics student, as noted on my user page. Probably you've been confused by the comment by Vb immediately above mine, which I suppose was signed with a ~ too many, displaying only the time in the signature. BTW I fixed that. -- Army1987 – Deeds, not words. 15:00, 21 January 2009 (UTC)
As for the lead, I don't think that factual accuracy and clarity are incompatible goals. While it's true that very few people know what the word field means, commutativity and associativity of addition and multiplication etc. are taught in grade schools. It shouldn't be impossible to write a lead which doesn't contain factual inaccuracies and yet can be understood by anyone in the last year of high school and also by sufficiently bright younger people. -- Army1987 – Deeds, not words. 15:16, 21 January 2009 (UTC)
Well, accuracy and clarity are not incompatible goals overall, over the span of 3-4 sentences they can be. Most high school students I have taught are much more comfortable with the concept of a vector as a pair of numbers then with the terms commutativity and associativity. While these are often taught in grade school, and again in middle school, and again in high school algebra, it doesn't exactly prepare students for the concept of "numbers" that are more general then the complex numbers. I think the goal for the lead should be to get a the idea across, and later in the article (say the definition section) we can discuss its more general formulations. Thenub314 (talk) 15:45, 21 January 2009 (UTC)
I agree with Thenub here. For what it's worth, as an undergrad math student I made a living tutoring people, and as a grad student I was a TA and taught courses in calculus, algebra, and discrete math, among other things, so I too have had some opportunities to learn what sorts of explanations actually work for people. Looie496 (talk) 17:55, 21 January 2009 (UTC)
### Concrete proposals
OK, so now everybody has given his/her opinion. In order to get it back on a more concrete track, may I propose the following procedure: everybody interested writes a lead section (1st paragraph only) and puts its here. Then we can see and discuss advantages of the drafts.
Here is my take (which is the current version) Jakob.scholbach (talk) 16:06, 21 January 2009 (UTC)
A vector space is a mathematical structure formed by a collection of objects, called vectors, that may be added and scaled, or multiplied by numbers. For example, physical forces acting on some body are vectors: any two forces can be added to yield a third, and the multiplication of a force vector by a real factor—also called scalar—is another force vector. General vector spaces are modelled on this and other examples such as geometrical vectors in that the operations of addition and scaling or scalar multiplication have to satisfy certain requirements that embody essential features of these examples. In addition to scaling by real numbers, vectors and vector spaces with multiplication by complex or rational numbers, or scalars in even more general mathematical fields are used.
Jakob.scholbach (talk) 16:06, 21 January 2009 (UTC)
A vector space is a mathematical structure formed by a collection of objects called vectors, along with two operations called vector addition and scalar multiplication. Vector spaces are a primary topic of the branch of mathematics called linear algebra, and they have many applications in mathematics, especially in mathematical physics. The most basic example of a vector space is the set of "displacements" in N-dimensional Euclidean space. Intuitively, a Euclidean displacement vector is often thought of as an arrow with a given direction and length. Addition of displacement vectors is done by placing them end-to-end, with the vector sum being a vector that points from the beginning of the first vector to the end of the second vector. Scalar multiplication is done by altering the length of a vector while keeping its direction the same. Many of the properties of N-dimensional Euclidean vector spaces generalize to vector spaces based on other number systems, or to infinite dimensional vector spaces whose elements are functions.
Looie496 (talk) 17:48, 21 January 2009 (UTC)
Places where confusion arises in the lead:
• The sentence beginning "for example" - readers unfamiliar with mathematics will be confused by this accumulation of terms
• Perhaps a simple definition could be added before the precise mathematical definition, one that would be more accessible to non-mathematicians.
• The "history" paragraph seems to interrupt the discussion of vectors
• The relationship between "collection of objects" and "physical forces", both described as vectors, is unclear.
These suggestions come from my writing class, which consists of college sophomores, juniors, and seniors from across the disciplines. Hope they help! Awadewit (talk) 17:44, 21 January 2009 (UTC)
Thanks! As you see, we are in the middle of the discussion. We'll use your hints. Jakob.scholbach (talk) 22:09, 21 January 2009 (UTC)
1) OK, 2) Hm. Currently (as previously) a pretty vague "definition" is given. Do you think "objects that may be added together and multiplied ("scaled") by numbers" is still too difficult to grasp? 3) OK. 4) OK, that was a mis-wording (the objects in the collection are vectors, the collection of vectors is the vector space). Is this clearer now? Jakob.scholbach (talk) 23:22, 22 January 2009 (UTC)
My class thought the new version was a dramatic improvement, in particular the phrase you have highlighted above - "objects that may be added....". Awadewit (talk) 20:31, 26 January 2009 (UTC)
## "A vector space is a set"
Out of curiosity, haven't vector spaces as proper classes been considered in the literature? GeometryGirl (talk) 13:44, 22 January 2009 (UTC)
I haven't come across it but I would be surprised if it didn't exist. If you ever run across a good reference it might be nice to include it in the generalizations section. Thenub314 (talk) 14:49, 22 January 2009 (UTC)
To support the ongoing FAC process, a few comments on the lead:
• Could shorten first paragraph by taking out the in-line explanation of what a field is: it is already wikilinked and a reader not familiar with the concept is not likely to learn yet another new definition while reading the lead;
• Yes, I think the same. I have shrunk that. Jakob.scholbach (talk) 23:17, 22 January 2009 (UTC)
• Euclidean vectors vs geometrical vectors: what is the intended difference? Current text appears to equate Euclidean vector with uses in physics, which is not really right. Should probably combine the terms geometric and Euclidean together (using only one of them) and then say that one very important use of these is representing forces in physics.
• To be honest, I don't know what to do about Euclidean vectors. Physicists seem to insist on them, mathematically they have little or no importance (at least they are hardly called like that). I tried to reword it to make clear that, in essence, the same concept is used both in physics and geometry. Jakob.scholbach (talk) 23:17, 22 January 2009 (UTC)
We had a long argument about them on Talk:Euclidean vector. If one accepts that a "Euclidean vector" is the same thing as a "contravariant vector", then we had a long discussion starting about here, and my conclusion was that contravariant vectors are tangent vectors. There was a moment when I was convinced they were something else, but I changed my mind later (see my last comment under here). Ozob (talk) 02:00, 23 January 2009 (UTC)
We physicists don't usually call them Euclidean vectors either. I don't think thatn-dimensional Euclidean vector space means anything more specific than any n-dimensional vector space over R with a positive-definite inner product. If one wants to specify vectors acting on the particular Euclidean affine space used in non-relativistic physics to model physical space, one would just say "space vectors", "spatial vectors", "three-vectors" or stuff like that. But people on Talk:Euclidean vector seem to think otherwise, and I got tired of arguing. -- Army1987 – Deeds, not words. 18:10, 23 January 2009 (UTC)
• The end of the first paragraph: "in that the axiom of vector addition and scalar multiplication embody essential features of these examples" is not very helpful and repeats the point. Could be cut to make the text mass lighter.
• Linear flavor: what does this mean? Some could call this a circular reference...
• Second paragraph should be split: it is not coherent as it begins with history (which should find its way in the lead) but ends with discussion of dimension.
• Could it be more comprehensible to try to define dimension (finite, at least) vaguely in terms of "independent directions" existing in the space (technically this would be the maximal number of linearly independent vectors) rather than with "number of scalars needed to specify a vector" (technically size of minimal set of generators)? The reader at this stage does not know how a vector can be specified using a list of scalars (once a base is given) but the intuition about independent directions could be provided with the list of one direction in a line, two in plane and three in space.
• Convergence questions is not very clear unless you know what's meant already. Could someting like Analytical problems call for the ability to decide if a sequence of vectors converges to a given vector provide more flavour without adding much text?
• Now get the impression that among topological vector spaces Banach and Hilbert spaces are particularly complicated, a viewpoint I would not accept. Intention is presumably to claim that these are particularly important types of TVS, which is surely right.
• OK. Reworded (and moved the footnote there). Jakob.scholbach (talk) 23:17, 22 January 2009 (UTC)
• Applications section is strangely skewed. It is true that given the almost ubiquitous applications of vector spaces both within mathematics and in other disciplines, it is hard to write a balanced paragraph. But singling out Fourier analysis looks unwarranted. Differential equations make sense, in that they were instrumental for the development of topological vector spaces. Local linearisation of manifolds may be a tad too technical as the other example. Systems of linear equations? High school background should make these something to relate to.
• Do you mean that the article is skewed or that the lead is skewed? If the article is OK, then the lead has but to sum up the article, so a word about Fourier and friends seems logical? Systems of linear equations are now mentioned.
Stca74 (talk) 15:03, 22 January 2009 (UTC)
## Using bullets for scalar multiplication
Don't you think that writing The product of a vector v and scalar a is denoted av, but then denoting it a · v in the rest of the article, can be confusing? Is the reader going to understand they refer to the same thing? Also, in equivalently 2 · v is the sum w + w, why the same vector should be referred to as v on the LHS and as w on the RHS, and why there shouldn't be a {{nowrap begin}}/{{nowrap end}} around the w + w., as there is one around similar such expressions in the same paragraph? And why the word ordered should be hidden by a link such as pair, in flagrant violation of WP:EGG? -- Army1987 – Deeds, not words. 14:26, 24 January 2009 (UTC)
I guess you refer to my reverting your edit. Sorry, I had not realized these changes, only the removing of the dots. (I did watch the diff, but somehow missed them). I have reinstated your points (thanks for catching the 2*v = w+w, in particular) and put a notice that the product may also be denoted with a dot. I think points like rv = (rx, ry) could be confusing to some readers. Jakob.scholbach (talk) 15:43, 24 January 2009 (UTC)
In view of some of the problems people are having with the lead over at the FAC page, I thought I'd put something down here to see if this is more along the lines of what they want. I'm thinking that what is desired is that at least the first paragraph be some layman's terms way of describing what vector spaces over the reals are. Delving into other fields and such could then be left to the later paragraphs of the lead. So my question is, is the following the type of content that the opposition at the FAC page is looking for (clearly, the prose itself is quite lacking)?
A vector space is a mathematical structure that, in its simplest form, can be used to track one or more quantities. In this way, they generalize the real numbers, which can be used to track one quantity as in "I am 5.2 km down the road from your house" or "I am missing 1.3 cups of sugar for this cake recipe" (-1.3 cups of sugar). An element of a vector space (called a vector) could represent a position using three distances such as "I am 2.3 km east, 1.1 km north from you and 100 m below ground" (which can be represented as a triple of real numbers (2.3, 1.1, -100) ), or it could represent how much sugar and flour one requires as in "These cupcakes need 1 cup of sugar and 3.25 cups of flour" (which can be represented as a pair of real numbers (1, 3.25) ). Like real numbers, vectors in a vector space can be scaled and added together. In other words, one can speak of multiplying a vector by a real number, as in "I want to make 2.5 times as many cupcakes, so I will need 2.5 cups of sugar and 8.125 cups of flour" (written as 2.5 · (1, 3.25) = (2.5, 8.125) ), and one can speak of adding two vectors together, as in "This cake asks for 1.5 cups of sugar and 2.75 cups of flour, so in total I will need 2.5 cups of sugar and 6 cups flour" (written as (1, 3.25) + (1.5, 2.75) = (2.5, 6) ). From a mathematical point of view, the specific quantities a vector represents are immaterial so that only the number of quantities matters. For this reason, the mathematical structure of a vector space is determined by the number of quantities it tracks (called the dimension of the vector space).
One could then go on to say that "Mathematicians have generalized certain properties of the real numbers to invent the concept of a "field" ..." etc.
Now, I realize this is a rather poorly written paragraph, but in particular it seems hard to clearly describe what is going on without all the examples. Though perhaps they could be relegated to the "motivation" section. Also, for a mathematician, this is probably a non-ideal beginning since a mathematician prefers to say what something is before describing what it can do. However, it seems like a compromise is necessary. I hope what I've written can lead to some progress on the issue. Cheers. RobHar (talk) 16:13, 24 January 2009 (UTC)
Thanks for your suggestion. Frankly, I would be somewhat unhappy to have such a paragraph in the lead section. It contradicts the credo (or guideline, if you want) that working out detailed examples should be avoided. Also, we are not writing a textbook (or cookbook :), sorry I couldn't resist). I like "[v.sp.] can be used to track one or more quantities." and we could perhaps integrate that to the lead. I'm repeating myself, but we can not explain the whole concept from scratch in the lead section of the article. This would be totally unbalanced (also contradicting some guideline). If anywhere, we can explain it with this level of detail in the body of the text. But, I think even there it is inappropriate to do it this way.
I'd like to put here, for comparison, the relevant lead section paragraph of group (mathematics), which is a featured article whose accessibility has been validated by lay readers. It reads
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and operation must satisfy a few conditions called group axioms, namely associativity, identity and invertibility. While these are familiar from many mathematical structures, such as number systems—for example, the integers endowed with the addition operation form a group—the formulation of the axioms is detached from the concrete nature of the group and its operation.
I should say I'm probably biased, because I contributed to that, but I think it has the right spirit of succinctly picking a simple key example and alluding to the concrete definition with its motivation/background/... which comes in the body. There will be many readers who will only fully understand the "integers and addition" thing. So what? We can not, for example (Notice that there are differences: e.g., group axioms are fewer. We should not mention the axioms of vsp in the lead).
Another example that comes to my mind: if you want to write the lead section for plane, say, you would not be able to talk in detail about explaining basics of aerodynamics, but perhaps just write that "Aircrafts often rely on carved wings, creating a difference in air pressure above and below the wings". You would and could not explain what pressure means, why moving air creates pressure differences etc. I think we have to face the reality that certain concepts can not be explained from scratch in one paragraph. Doing our best to educate the reader with the text body is our duty, and we should excel in it. However, putting everything into the lead is simply not going to work. Jakob.scholbach (talk) 16:59, 24 January 2009 (UTC)
RobHar's suggestion is well-meaning but totally contrary to WP:LEAD ("The lead serves both as an introduction to the article below and as a short, independent summary of the important aspects of the article's topic.") and the principle that Wikipedia is not a textbook. Geometry guy 20:39, 24 January 2009 (UTC)
Using the Group (mathematics) article's lead as a basis (pun not intended):
In mathematics, a vector space is an algebraic structure consisting of a set of vectors together with two operations, addition and scaling. These vectors track one or more quantities, such as displacement (the distance travelled and in which direction), or force. To qualify as a vector space, the set and operations must satisfy a few conditions called axioms. These axioms generalize the properties of Euclidean space (e.g. the plane, an idealized flat surface), but their formulation is detached from the concrete nature of any given vector space. Concepts like length and angle may be undefined or counter-intuitive in certain vector spaces.
I think this a bit too abstract. Also Euclidean space is too technical. Is there anything else that vs's generalize that is less technical? Alksentrs (talk) 17:55, 24 January 2009 (UTC)
Let me point out that the Euclidean vector article is quite nice, with minimal assumptions of background on the part of the reader. It might be helpful to direct readers with a weak background there—a reader who has read that article should be in a much better position to understand this one. Looie496 (talk) 18:15, 24 January 2009 (UTC)
## Three elementary consequences of the definition need to be given explicitly
I think that three important consequences of the definition need to be stated. Namely that for all scalars a, and vectors u, the following hold:
1. a0 = 0
2. 0u = 0
3. (−1)u = −u
Note that 2 is expressed in words, but I think it would be good to express it as a formula.
Paul August 17:41, 24 January 2009 (UTC)
I'm not sure. You are right that it is somewhat important to know these facts, but we can not (for space reasons) put everything that is important. Also, we have to maintain balance of elementary and advanced material. That said, I would propose adding a precise citation to the relevant paragraph, pointing to a book or so that has these (and further) elementary consequences. Jakob.scholbach (talk) 15:55, 25 January 2009 (UTC)
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2016-07-29 11:45:27
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https://www.physicsforums.com/threads/first-countable-spaces-and-metric-spaces.511838/
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# Homework Help: First countable spaces and metric spaces.
1. Jul 4, 2011
### alemsalem
1. The problem statement, all variables and given/known data
Show That every metric space is first countable. Hence show that every SUBSET of a metric space is the intersection of a countable family of open sets.
2. Relevant equations
no equation
3. The attempt at a solution
its easy to show that it is first countable, because for every point in the space there is the set of rational open balls which are included in every other open set.
but the second part of the question is confusing:
how can every subset be an intersection of a countable family? we only know that at every point there is a countable family but. there maybe an uncountable number of point..
Thanks for the help, i've been staring at the question for two hours..
by the way this problem is from P. Szekeres chapter 10 problem 10.9
Last edited: Jul 4, 2011
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2018-10-22 23:49:51
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https://tex.stackexchange.com/questions/227065/how-to-set-default-footnote-linespacing
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# How to set default footnote linespacing? [duplicate]
How to set default footnote linespacing? Now for each footnote I have to write
\footnote{\onehalfspacing ...}
But I would like to set the line spacing once.
I guess its possible to use \renewcommand{\footnote}{...}. But I don't know how to redefine the command.
• All packages that make line spacing changes easy take care, that this does not happen in footnotes. There is good reason for that. Are you sure, you want to alter this? – Johannes_B Feb 7 '15 at 18:45
• @Johannes_B Yes, I am sure, because this is a strange requirement for documents in my university. – niekas Feb 7 '15 at 18:46
• So, the simplest solution would be to not use package setspace and change it by hand for the whole and complete document. Depending on the pointsize, you can set in the preamble \setstretch{1.25} (10pt), 1,213 for 11pt or 1.241 for 12pt basesize. – Johannes_B Feb 7 '15 at 18:46
• @Johannes_B But \setstretch command is imported from setspace package. I am defining document class, so it would be best to set default footnote line spacing in it, not in the main document. – niekas Feb 7 '15 at 19:02
• I thought this might be a declaration for the whole document. University specs are very strange, just as if they never noticed that typewritters have gone. Try Can I redefine a command to contain itself? – Johannes_B Feb 7 '15 at 19:06
You can do something very similar to Can I redefine a command to contain itself?
\documentclass{article}
\usepackage{blindtext}
\usepackage{letltxmacro}
\begin{document}
\blindtext
\footnote{\blindtext}
\LetLtxMacro{\niekasFootnote}{\footnote}
\makeatletter
\renewcommand{\footnote}[1]{\niekasFootnote{\def\baselinestretch{1.25}\@currsize#1}}
\makeatother
\footnote{\blindtext}
\end{document}
As you will place that in a class file, the \makeatletter \makeatother combo can be omitted. The definition comes form the source of package setspace. If this package is loaded anyways, you can use onehalfspacing (or any other of the spacing comands defined by setspace). Advantage: Those commands are aware of the current font size, since different sizes need different spacing to look right.
• Why do you need \@currsize? Omitting it does not require \makeatletter and \makeatother commands anymore. – niekas Feb 7 '15 at 19:23
• @niekas That is the originaldefinition in setspace. If you load setspace anyway, you can savely use onehalfspacing which is aware of the type size – Johannes_B Feb 7 '15 at 19:24
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2021-06-20 10:16:07
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https://chemistry.stackexchange.com/questions/76487/how-can-i-calculate-the-standard-reaction-enthalpy-if-i-lack-one-formation-entha
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How can I calculate the standard reaction enthalpy if I lack one formation enthalpy?
I have this reaction, for which I need the standard enthalpy of reaction:
$$\ce{CuO (s) + H2S (g) \rightarrow CuS (s) + H2O (g)}$$
This is the data from the NIST web book (standard formation enthalpies, $\mathrm{kJ}\cdot\mathrm{mol}^{-1}$):
• $\ce{CuO (s)}$: -156.60
• $\ce{H2S (g)}$: -20.5
• $\ce{H2O (g)}$ -241.83
There are no data tabulated for $\ce{CuS (s)}$, so how can I compute the standard reaction enthalpy lacking this data?
• To put it bluntly, you can't. – Ivan Neretin Jun 21 '17 at 21:39
The standard enthalpy of formation of $\ce{CuS}\,\mathrm{(s)}$ is -48.4 $\mathrm{kJ}\cdot\mathrm{mol}^{-1}$. See this reference for your requested data and more.
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2020-07-06 18:27:06
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http://mymathforum.com/algebra/42587-inequality.html
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My Math Forum An inequality
Algebra Pre-Algebra and Basic Algebra Math Forum
April 5th, 2014, 08:17 AM #1 Senior Member Joined: Apr 2013 Posts: 425 Thanks: 24 An inequality To solve the inequality $\displaystyle 3ix^2+4x-5i\leq 0$ where $\displaystyle i^2=-1$.
April 5th, 2014, 08:52 AM #2 Senior Member Joined: Apr 2014 From: Greater London, England, UK Posts: 320 Thanks: 156 Math Focus: Abstract algebra $\displaystyle x\,=\,-\sqrt{\frac53}$.
April 5th, 2014, 06:35 PM #3 Math Team Joined: Dec 2013 From: Colombia Posts: 7,671 Thanks: 2651 Math Focus: Mainly analysis and algebra I couldn't disagree more. To me the inequality is meaningless because the complex plane has no ordering. Should there be some modulus symbols somewhere? Is $x$ supposed to be real or complex?
April 5th, 2014, 07:03 PM #4 Global Moderator Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,950 Thanks: 1141 Math Focus: Elementary mathematics and beyond Try subbing in Olinguito's answer.
April 5th, 2014, 08:26 PM #5 Math Team Joined: Dec 2013 From: Colombia Posts: 7,671 Thanks: 2651 Math Focus: Mainly analysis and algebra I know where that came from, and yes, by eliminating the imaginary part of the equation it makes some sort of sense of the inequality. But I would suggest that it's a poorly set question if that is the answer that is being sought.
April 5th, 2014, 09:26 PM #6 Senior Member Joined: Apr 2014 From: Greater London, England, UK Posts: 320 Thanks: 156 Math Focus: Abstract algebra Perhaps it was a trick question.
April 5th, 2014, 10:15 PM #7
Senior Member
Joined: Apr 2013
Posts: 425
Thanks: 24
Quote:
Originally Posted by Olinguito $\displaystyle x\,=\,-\sqrt{\frac53}$.
There are many more values of $\displaystyle x$ that verifies the inequality.
They are,therefore, all values of $\displaystyle x$ that verifies the inequality?
April 5th, 2014, 11:57 PM #8 Senior Member Joined: Apr 2014 From: Greater London, England, UK Posts: 320 Thanks: 156 Math Focus: Abstract algebra If you allow $x$ to be complex-valued, you could also have $$x\,=\,\frac{\pm\sqrt{11}+2i}3$$ By the way, was it meant to be a trick question – i.e. was it meant to provoke the reaction “this inquality is meaningless”? If it was, it was perfect! Had you posted it a few days earlier it would have even made an excellent April fool’s joke. PS: I’d like to point out that just because an inequality contains complex numbers does not mean that it must be meaningless. For example, the interior of unit circle can be described by $\displaystyle z\overline z<1$. If this inequality makes sense, despite the presence of complex variables, I don’t see why we should not try to make sense of the OP’s inequality. Last edited by Olinguito; April 6th, 2014 at 12:29 AM.
April 6th, 2014, 12:47 AM #9 Senior Member Joined: Mar 2014 Posts: 112 Thanks: 8 Olinguito is correct because 3ix² - 5i = 0 and 4x ≤ 0. The key to solving this problem correctly is to see how to get rid of the imaginary numbers to solve such problem. There could be more solutions but I won't go through all of them.
April 6th, 2014, 06:09 AM #10
Senior Member
Joined: Apr 2013
Posts: 425
Thanks: 24
Quote:
Originally Posted by Olinguito If you allow $x$ to be complex-valued, you could also have $$x\,=\,\frac{\pm\sqrt{11}+2i}3$$
Solving of the inequality $\displaystyle 3ix^2+4x-5i\leq 0$ where $\displaystyle i^2=-1$ is simple if it is considered an equivalent equation $\displaystyle 3ix^2+4x-5i=a$ where $\displaystyle i^2=-1$ and $\displaystyle a\in R,a\leq 0$.
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Contact - Home - Forums - Cryptocurrency Forum - Top
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2019-07-22 22:20:41
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https://blog.shahinrostami.com/2017/11/docker-quick-guide/
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Docker is a fantastic platform which has changed my workflow for the better. Before Docker, I was using VMWare images and linked-clones to create isolated environments for my experiments. Sharing these was not easy – the file sizes were too large and not everyone has a VMWare license. Performance wasn’t great either, and VMWare updates would sometimes cause problems with the image. Before VMWare, I was using MATLAB and directly sharing the scripts for my experiments. This was an unpleasant experience, as each of the many versions of MATLAB (e.g. R2011a, R2011b, R2012a, R2012b, etc.) would break some functionality from one of the toolboxes. Again, not everyone could afford a MATLAB license, and I only had mine as part of my institution.
With Docker, I’ve moved to using Python’s scientific stack and libraries such as TensorFlow and Keras. It took a while to get used to living without the MATLAB IDE, but there was a similar IDE called Spyder, which is a Scientific Python Development EnviRonment. However, I’ve gotten over the MATLAB IDE since discovering Jupyter. With all these technologies, I can share an environment identical to the one I executed my experiments in, and allow a colleague or reviewer to do the same. My students can also get up and running with the environments needed by my taught units without having to configure everything themselves. The only difficulty I’ve encountered so far is when someone has a Windows installation which doesn’t support Docker for Windows, as Docker Toolbox isn’t always a smooth experience.
I’ve created this video as a brief introduction to Docker, complete with instructions on how to install Docker on macOS, Windows 10, and Ubuntu 16.04:
The aim is to get someone started and tinkering with Docker quickly – I hope it’s helpful.
## Common Problems
### Kernel appears to have died
If your kernel dies when training a network using TensorFlow, then you might need to increase the amount of RAM allocated to Docker. You can do this in the Docker Preferences:
### I have Windows 10 Pro/Enterprise/Education, but the installer says I don’t meet the minimum requirements
You need to be on Windows 10 build 10586 or newer. You can check your Windows build number by clicking the start-bar and winver, and pressing the enter key:
Author
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2019-05-21 07:09:37
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https://brilliant.org/problems/fun-with-maths-3/
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# Fun with Maths
Algebra Level 4
If the roots of $$10x^3- cx^2-54x-27=0$$ are in harmonic progression then find $$c$$.
×
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2017-09-26 14:51:27
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https://nigerianscholars.com/past-questions/math/question/602329/
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» » » Some values of the linear function f are shown in the table above. Which of t...
Some values of the linear function f are shown in the table above. Which of t...
Question
Some values of the linear function f are shown in the table above. Which of the following defines f ?
Options
A)
f(x) = 2x + 3
B)
f(x) = 3x + 2
C)
f(x) = 4x + 1
D)
f(x) = 5x
Choice C is correct. Because f is a linear function of x, the equation f(x) = mx + b, where m and b are constants, can be used to define the relationship between x and f(x). In this equation, m represents the increase in the value of f(x) for every increase in the value of x by 1. From the table, it can be determined that the value of f(x) increases by 8 for every increase in the value of x by 2. In other words, for the function f the value of m is $$\frac{8}{2}$$, or 4. The value of b can be found by substituting the values of x and f(x) from any row of the table and the value of m into the equation f(x) = mx + b and solving for b. For example, using x = 1, f(x) = 5, and m = 4 yields 5 = 4(1) + b. Solving for b yields b = 1. Therefore, the equation defining the function f can be written in the form f(x) = 4x + 1.
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2023-02-04 15:43:35
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https://zbmath.org/?q=an:0757.49012
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# zbMATH — the first resource for mathematics
Integral representation of nonconvex functionals defined on measures. (English) Zbl 0757.49012
The paper deals with an integral representation problem for nonconvex functionals defined on the space $$M(\Omega;\mathbb{R}^ n)$$ of the $$\mathbb{R}^ n$$-valued measures with bounded variation on $$\Omega$$. More precisely given a separable locally convex metric space $$\Omega$$ and a functional $$F:M(\Omega;\mathbb{R}^ n)\to[0,+\infty]$$ it is proved that if $$F$$ verifies the following assumptions: (i) $$F$$ is additive (i.e. $$F(\lambda+\mu)=F(\lambda)+F(\mu)$$ whenever $$\lambda,\mu\in M(\Omega;\mathbb{R}^ n)$$ and $$\lambda$$ is singular with respect to $$\mu)$$, (ii) $$F$$ is sequentially weakly* lower semicontinuous on $$M(\Omega;\mathbb{R}^ n)$$ then there exist a non-atomic positive measure $$\mu$$ in $$M(\Omega;\mathbb{R}^ n)$$ and three Borel functions $$f,\varphi,g:\Omega\times\mathbb{R}^ n\to[0,+\infty]$$ verifying suitable conditions such that the following integral representation formula holds for every $$\lambda\in M(\Omega;\mathbb{R}^ n)$$: $F(\lambda)=\int_ \Omega f\left(x,{d\lambda\over d\mu}\right)d\mu+\int_{\Omega\backslash A_ \lambda}\varphi\left(x,{d\lambda^ 2\over d|\lambda^ s|}\right)d|\lambda^ s|+\int_{A_ \lambda}g(x,\lambda(\{ x\}))d\#, \tag{*}$ where $${d\lambda\over d\mu}$$ denotes the Radon- Nikodym derivative of $$\lambda$$ with respect to $$\mu$$, $${d\lambda^ s\over d|\lambda^ s|}$$ the one of $$\lambda^ s$$ with respect to $$|\lambda^ s|$$, $$A_ \lambda$$ is the set of the atoms of $$\lambda$$ and $$\#$$ is the counting measure on $$\Omega$$. The uniqueness of the representation of $$F$$ in the form $$(*)$$ is also discussed.
The above result extends to the nonconvex case analogous integral representation theorems already existing in literature but relative to convex functionals verifying (i) and (ii).
It is also proved that the necessary conditions on $$f$$, $$\varphi$$ and $$g$$ given by the previous theorem, together with a mild additional assumption, become sufficient in order to get a weak*-$$M(\Omega;\mathbb{R}^ n)$$ lower semicontinuity result for the functional in $$(*)$$. This result extends those already existing in literature.
Finally some examples are discussed, in particular the last one proves that the assumptions of the abover mentioned lower semicontinuity result are sharp.
##### MSC:
49J45 Methods involving semicontinuity and convergence; relaxation
Full Text:
##### References:
[1] Ambrosio, L.; Buttazzo, G., Weak lower semicontinuous envelope of functionals defined on a space of measures, Ann. Mat. Pura Appl., Vol. 150, 311-340, (1988) · Zbl 0648.49009 [2] Bouchitté, G., Représentation intégrale de fonctionnelles convexes sur un espace de mesures, Ann. Univ. Ferrara, Vol. 33, 113-156, (1987) · Zbl 0721.49041 [3] Bouchitté, G.; Buttazzo, G., New lower semicontinuity results for non convex functionals defined on measures, Nonlinear Anal., Vol. 15, 7, 679-692, (1990) · Zbl 0736.49007 [4] Bouchitté, G.; Valadier, M., Integral representation of convex functionals on a space of measures, J. Funct. Anal., Vol. 80, 398-420, (1988) · Zbl 0662.46009 [5] Bouchitté, G.; Valadier, M., Multifonctions s.c.i. et régularisée s.c.i. essentielle. Proceedings “congrès franco-québécois d’analyse non linéaire appliquée”, Perpignan, June 22-26, 1987, Analyse Non linéaire - Contribution en l’Honneur de J. J. Moreau, 123-149, (1989), Bordas Paris · Zbl 0704.49017 [6] Buttazzo, G., Semicontinuity, relaxation and integral representation in the calculus of variations, Pitman Res. Notes Math. Ser., Vol. 207, (1989), Longman Harlow · Zbl 0669.49005 [7] Buttazzo, G.; Dal Maso, G., On nemyckii operators and integral representation of local functionals, Rend. Mat., Vol. 3, 491-509, (1983) · Zbl 0536.47027 [8] de Giorgi, E.; Ambrosio, L.; Buttazzo, G., Integral representation and relaxation for functionals defined on measures, Atti Accad. Naz. Lincei Rend. CI Sci. Fis. Mat. Natur., Vol. 81, 7-13, (1987) · Zbl 0713.49018 [9] Demengel, F.; Теmam, R., Convex functions of measures and applications, Indiana Univ. Math. J., 33, 673-709, (1984) · Zbl 0581.46036 [10] Dunford, N.; Schwartz, J. T., Linear operators, (1957), Interscience Publishers Inc New York [11] Goffman, C.; Serrín, J., Sublinear functions of measures and variational integrals, Duke Math. J., Vol. 31, 159-178, (1964) · Zbl 0123.09804 [12] Hiai, F., Representation of additive functionals on vector valued normed Köthe spaces, Kodai Math. J., Vol. 2, 300-313, (1979) · Zbl 0431.46025 [13] Ioffe, A. D., On lower semicontinuity of integral functionals I. II, SIAM J. Control Optim., Vol. 15, 521-538, (1977), nd 991-1000 · Zbl 0361.46037 [14] Rockafellar, R. T., Integrals which are convex functionals I. II, Pacific J. Math., Vol. 39, 439-469, (1971) · Zbl 0236.46031 [15] Rockafellar, R. T., Convex analysis, (1972), Princeton University Press Princeton · Zbl 0224.49003 [16] Valadier, M., Closedness in the weak topology of the dual pair L^{1}, C, J. Math. Anal. Appli., Vol. 69, 17-34, (1979) · Zbl 0412.46040
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.
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2021-10-21 05:32:10
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https://www.quizover.com/physics-ap/course/24-3-the-electromagnetic-spectrum-by-openstax?page=7
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# 24.3 The electromagnetic spectrum (Page 8/33)
Page 8 / 33
Living things—plants and animals—have evolved to utilize and respond to parts of the electromagnetic spectrum they are embedded in. Visible light is the most predominant and we enjoy the beauty of nature through visible light. Plants are more selective. Photosynthesis makes use of parts of the visible spectrum to make sugars.
## Integrated concept problem: correcting vision with lasers
During laser vision correction, a brief burst of 193-nm ultraviolet light is projected onto the cornea of a patient. It makes a spot 0.80 mm in diameter and evaporates a layer of cornea $0\text{.}\text{30}\phantom{\rule{0.25em}{0ex}}\mu \text{m}$ thick. Calculate the energy absorbed, assuming the corneal tissue has the same properties as water; it is initially at $\text{34º}\text{C}$ . Assume the evaporated tissue leaves at a temperature of $\text{100º}\text{C}$ .
Strategy
The energy from the laser light goes toward raising the temperature of the tissue and also toward evaporating it. Thus we have two amounts of heat to add together. Also, we need to find the mass of corneal tissue involved.
Solution
To figure out the heat required to raise the temperature of the tissue to $\text{100º}\text{C}$ , we can apply concepts of thermal energy. We know that
$\text{Q}=\text{mc}\Delta T,$
where Q is the heat required to raise the temperature, $\Delta T$ is the desired change in temperature, $m$ is the mass of tissue to be heated, and $c$ is the specific heat of water equal to 4186 J/kg/K.
Without knowing the mass $m$ at this point, we have
$Q=m\left(\text{4186 J/kg/K}\right)\left(\text{100º}\text{C}–\text{34º}\text{C}\right)=m\left(\text{276,276 J/kg}\right)=m\left(\text{276 kJ/kg}\right).$
The latent heat of vaporization of water is 2256 kJ/kg, so that the energy needed to evaporate mass $m$ is
${Q}_{\text{v}}={\mathrm{mL}}_{\text{v}}=m\left(\text{2256 kJ/kg}\right).$
To find the mass $m$ , we use the equation $\rho =m/\text{V}$ , where $\rho$ is the density of the tissue and $\text{V}$ is its volume. For this case,
$\begin{array}{lll}m& =& \rho \text{V}\\ & =& {\text{(1000 kg/m}}^{3}\right)\left(\text{area}×\text{thickness}{\text{(m}}^{3}\text{))}\\ & =& \text{(1000 kg/}{\text{m}}^{3}\right)\left(\pi \left(0.80×{\text{10}}^{–3}\phantom{\rule{0.25em}{0ex}}\text{m}{\right)}^{2}/4\right)\left(0\text{.}\text{30}×{\text{10}}^{–6}\phantom{\rule{0.25em}{0ex}}\text{m}\right)\\ & =& 0.151×{\text{10}}^{–9}\phantom{\rule{0.25em}{0ex}}\text{kg.}\end{array}$
Therefore, the total energy absorbed by the tissue in the eye is the sum of $\text{Q}$ and ${\text{Q}}_{\text{v}}$ :
${\text{Q}}_{\text{tot}}=m\left(c\Delta \text{T}+{\text{L}}_{\text{v}}\right)=\left(0.151×{\text{10}}^{-9}\phantom{\rule{0.25em}{0ex}}\text{kg}\right)\left(\text{276 kJ/kg}+\text{2256 kJ/kg}\right)=\text{382}×{\text{10}}^{-9}\phantom{\rule{0.25em}{0ex}}\text{kJ}.$
Discussion
The lasers used for this eye surgery are excimer lasers, whose light is well absorbed by biological tissue. They evaporate rather than burn the tissue, and can be used for precision work. Most lasers used for this type of eye surgery have an average power rating of about one watt. For our example, if we assume that each laser burst from this pulsed laser lasts for 10 ns, and there are 400 bursts per second, then the average power is ${\text{Q}}_{\text{tot}}×\text{400}=\text{150 mW}$ .
Optics is the study of the behavior of visible light and other forms of electromagnetic waves. Optics falls into two distinct categories. When electromagnetic radiation, such as visible light, interacts with objects that are large compared with its wavelength, its motion can be represented by straight lines like rays. Ray optics is the study of such situations and includes lenses and mirrors.
When electromagnetic radiation interacts with objects about the same size as the wavelength or smaller, its wave nature becomes apparent. For example, observable detail is limited by the wavelength, and so visible light can never detect individual atoms, because they are so much smaller than its wavelength. Physical or wave optics is the study of such situations and includes all wave characteristics.
What is meant by dielectric charge?
what happens to the size of charge if the dielectric is changed?
omega= omega not +alpha t derivation
u have to derivate it respected to time ...and as w is the angular velocity uu will relace it with "thita × time""
Abrar
do to be peaceful with any body
the angle subtended at the center of sphere of radius r in steradian is equal to 4 pi how?
if for diatonic gas Cv =5R/2 then gamma is equal to 7/5 how?
Saeed
define variable velocity
displacement in easy way.
binding energy per nucleon
why God created humanity
Because HE needs someone to dominate the earth (Gen. 1:26)
Olorunfemi
Ali
Is the object in a conductor or an insulator? Justify your answer. whats the answer to this question? pls need help figure is given above
ok we can say body is electrically neutral ...conductor this quality is given to most metalls who have free electron in orbital d ...but human doesn't have ...so we re made from insulator or dielectric material ... furthermore, the menirals in our body like k, Fe , cu , zn
Abrar
when we face electric shock these elements work as a conductor that's why we got this shock
Abrar
how do i calculate the pressure on the base of a deposit if the deposit is moving with a linear aceleration
why electromagnetic induction is not used in room heater ?
room?
Abrar
What is position?
What is law of gravition
what is magnetism
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2018-09-23 16:15:28
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https://cloud.r-project.org/web/packages/protti/vignettes/data_analysis_single_dose_treatment_workflow.html
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# Introduction
This vignette will give you an overview of how you can analyse bottom-up proteomics data or LiP-MS data using protti. If you would like to analyse dose-response data please refer to the dose-response data analysis vignette. Before analysing your data make sure that it is of sufficient quality and that you do not have any outliers. To do this you can take a look at the quality control vignette.
protti includes several functions that make it easy for the user to analyse and interpret data from bottom-up proteomics or LiP-MS experiments. The R package includes functions for
• Quality control (see quality control vignette for more detailed information)
• Data preparation
• Median normalisation
• Data filtering
• Protein abundance calculation from precursor or peptide intensities
• Imputation of missing values
• Fetching of database information (ChEBI, GO, KEGG, MobiDB, UniProt)
• Calculation of sequence coverage
• Data analysis
• Statistical hypothesis tests
• Data visualisation
• Volcano plots
• Barcode plots (plots that show protein coverage and significant changes projected onto the protein sequence)
• Wood’s plots
• Profile plots
• Data interpretation
• Treatment enrichment (check if your hits are enriched with known targets)
• GO-term enrichment
• Network analysis (based on STRING database information)
• KEGG pathway enrichment
You can read more about specific functions and how to use them by calling e.g. ?normalise (for the normalise() function). Calling ? followed by the function name will display the function documentation and give you more detailed information about the function. This can be done for any of the functions included in the package.
This document will give you an overview of data preparation, data analysis, data visualisation and data interpretation functions included in protti. It will show you how they can be applied to your data. The examples in this file are run on a published LiP-MS data set. In the experiment a HeLa cell lysate was treated with different amounts of the drug rapamycin and the target of rapamycin (FKBP12) was successfully identified. For this vignette we are using a filtered version of the original data set where we include only the 10 μM treatment concentration and the control (untreated). For simplicity only 50 proteins are included in the data set.
The data set is produced from the output of Spectronaut™. However, if you have any other data such as DDA data that was searched with a different search engine you can still apply protti’s functions. Just make sure that your data frame contains tidy data. That means data should be contained in a long format (e.g. all sample names in one column) rather than a wide format (e.g. each sample name in its own column). You can easily achieve this by using the pivot_longer() function from the tidyr package. If you are unsure what your input data is supposed to look like, please use the create_synthetic_data() function and compare this to your data. You can also take a look at the input preparation vignette, there you will find all the necessary information on how to get your data into the correct format.
The input data should have a similar structure to this example:
Sample Replicate Peptide Condition log2(Intensity)
sample1 1 PEPTIDER treated 14
sample1 1 PEPTI treated 16
sample1 1 PEPTIDE treated 17
sample2 1 PEPTIDER untreated 15
sample2 1 PEPTI untreated 18
sample2 1 PEPTIDE untreated 12
# How to use protti to analyse your data
## Getting started
Before we can start analysing our data, we need to load the protti package. This is done by using the base R function library(). In addition, we are also loading the packages magrittr and dplyr. Both magrittr and dplyr are part of the tidyverse, a collection of packages that provide useful functionalities for data processing and visualisation. If you use many tidyverse packages in your workflow you can easily load all at once by calling library(tidyverse).
library(protti)
library(magrittr)
library(dplyr)
After having loaded the required packages we will load our data set into the R environment. In order to do this for your data set you can use the function read_protti(). This function is a wrapper around the fast fread() function from the data.table package and the clean_names() function from the janitor package. This will allow you to not only load your data into R very fast, but also to clean up the column names into lower snake case to make them more R-friendly. This will make it easier to remember them and to use them in your data analysis.
# To read in your own data you can use read_protti()
your_data <- read_protti(filename = "mydata/data.csv")
For this example we are going to use the rapamycin_10uM test data set included in protti. To read in the file we are simply going to use the utils function data().
data("rapamycin_10uM")
## Data preparation
### Log2 transformation, median normalisation and CV filtering
After inspecting the data and performing quality control (see quality control vignette for more information) we will now start to prepare the data for the analysis.
First, we remove decoy hits (used for false discovery rate estimation). Our example data set contains a column called eg_is_decoy that consists of logicals indicating whether or not the peptide is a decoy hit. To remove decoys we will use the dplyr function filter().
Next, we log2 transform our intensities, then normalise the data to the median value of all runs. To transform the intensities we use the dplyr function mutate() which creates a new column while maintaining the original column.
Note that we are also using the pipe operator %>% included in the R package magrittr. %>% takes the output of the preceding function and supplies it as the first argument of the following function. Using %>% makes code easier to read and follow.
For normalisation we are using the protti function normalise(). For this example we will use median normalisation (method = "median"). The function normalises intensities for each run to the median of all runs. This is only necessary if your search algorithm does not already median normalise your intensities. For the example data we have disabled median normalisation in Spectronaut, therefore we need to median normalise now. The formula for median normalisation is:
$median ~ normalised ~ intensity = intensity - median ( run ~ intensity ) + median ( global ~ intensity)$
To ensure that only good peptide measurements will be used for further analysis, we will also filter our data based on coefficients of variation (CV). In order to do this we are using the function filter_cv(). For this example we are retaining peptides with a CV < 25 % in at least one of the two conditions.
The CVs are calculated within the function according to the formula:
$CV = \frac{standard ~ deviation}{mean} * 100$
Note: The use of the filter_cv() function is optional. It might remove a lot of your data if your experiment was noisy. However, especially in these cases, the function will remove peptides with poor quality and should improve the result. It is very important that if you use this function you should not use the moderated t-test or proDA algorithm on your data for differential abundance estimation and significance testing. This likely will lead to an inflated false positive rate because it alters the distributional assumptions of these tests (Bourgon 2010).
data_normalised <- rapamycin_10uM %>%
filter(eg_is_decoy == FALSE) %>%
mutate(intensity_log2 = log2(fg_quantity)) %>%
normalise(sample = r_file_name,
intensity_log2 = intensity_log2,
method = "median")
data_filtered <- data_normalised %>%
filter_cv(grouping = eg_precursor_id,
condition = r_condition,
log2_intensity = intensity_log2,
cv_limit = 0.25,
min_conditions = 1)
### Remove non-proteotypic peptides
For LiP-MS analysis we commonly remove non-proteotypic peptides (i.e. peptides that could come from more than one protein). If you detect a change in non-proteotypic peptides it is not possible to clearly assign which protein it comes from and therefore which protein is affected by your treatment. If you are using the output from Spectronaut you will find a column called “pep_is_proteotypic”. This column contains logicals indicating whether your peptide is proteotypic or not.
To filter out non proteotypic peptides we are using the dplyr function filter().
data_filtered_proteotypic <- data_filtered %>%
filter(pep_is_proteotypic == TRUE)
### Fetching database information and assigning peptide types
In order to obtain more information about our identified proteins we are going to use the function fetch_uniprot() to download information from UniProt directly.
fetch_uniprot() uses a vector of UniProt IDs as its input. We produce this vector by using the base R function unique() which will extract all unique elements in the selected protein ID column. In this case we want to download the full protein name, gene IDs, GO terms associated with molecular function, StringDB IDs, information on known interacting proteins, location of the active site, location of binding sites, PDB entries, protein length and protein sequence. There are more options for columns to add (for more information on possible columns to add click here).
fetch_uniprot() returns a new data frame. In order to be able to merge this with our original data frame we have to rename the ID column to match the name of the protein ID column of our original data frame. To do this we use dplyr’s rename() function.
To merge the two data frames we use the dplyr function left_join(). We match the two data frames by the column “pg_protein_accessions”. By using left_join() we retain all rows from our original data frame while adding the columns from the data fame generated with fetch_uniprot().
Note: you can also directly join the UniProt data frame with your data without the need to rename its id column. You can specify in the by argument in left_join() that two columns are differently named.
Next, we would like to assign the trypticity of our peptides (i.e. if the peptides are fully-tryptic, semi-tryptic or non-tryptic). In order to do this we first need to define the peptide positions in the protein and find the preceding and following amino acids. To obtain this information we use the protti function find_peptide(). The output of this function can then be used in the function assign_peptide_type() which will add an additional column with the peptide trypticity information. By using the function calculate_sequence_coverage() we add an additional column to the data frame containing information on how much of the protein sequence we identified in our experiment.
uniprot_ids <- unique(data_filtered_proteotypic\$pg_protein_accessions)
uniprot <-
fetch_uniprot(
uniprot_ids = uniprot_ids,
columns = c(
"protein names",
"genes",
"go(molecular function)",
"database(String)",
"interactor",
"feature(ACTIVE SITE)",
"feature(BINDING SITE)",
"database(PDB)",
"length",
"sequence"
)
) %>%
rename(pg_protein_accessions = id)
data_filtered_uniprot <- data_filtered_proteotypic %>%
left_join(y = uniprot,
by = "pg_protein_accessions") %>%
find_peptide(protein_sequence = sequence,
peptide_sequence = pep_stripped_sequence) %>%
assign_peptide_type(aa_before = aa_before,
last_aa = last_aa,
aa_after = aa_after) %>%
calculate_sequence_coverage(protein_sequence = sequence,
peptides = pep_stripped_sequence)
With the qc_sequence_coverage() function, you check how sequence coverage is distributed over all proteins in the sample. Usually, the center of the distribution is low due to many proteins with poor coverage. For this small data set with only 40 proteins the sequence coverage is distributed relatively evenly.
qc_sequence_coverage(
data = data_filtered_uniprot,
protein_identifier = pg_protein_accessions,
coverage = coverage
)
## Data analysis
### Statistical hypothesis test
To test if there is a difference between the peptide abundances in our two conditions (i.e. rapamycin treated and untreated) we use a moderated t-test based on the limma R package.
Before the statistical hypothesis test we have to define the types of missing values present in our data set. We are going to use the function assign_missingness() which will return a column with information on the types of missingness we have in our data (i.e. complete, missing at random (MAR) or missing not at random (MNAR)). We use the default parameters of this function which assumes that missing values are MAR when the conditions are at least 70 % complete (adjusted downward). Missing values are assumed to be MNAR when less than 20 % of values are present (adjusted_downward) in one condition if the other condition is complete. If not “complete” all other comparisons are label as NA. If imputation is performed, these are the comparisons that will not be imputed. The type of missingness assigned to a comparison does not have any influence on the statistical test. However, by default (can be changed) comparisons with missingness NA are filtered out prior to p-value adjustment. This means that in addition to imputation the user can use missingness cutoffs also in order to define which comparisons are too incomplete to be trustworthy even if significant.
After assigning the types of data missingness we use the function calculate_diff_abundance(). By selecting method = t-test the function will perform a Welch’s t-test. There are also options included to perform a moderated t-test based on the R package limma or to detect differential abundances based on the algorithm implemented in the R package proDA. The algorithm used for proDA is based on a probabilistic dropout model which facilitates hypothesis testing (using a moderated t-test) while eliminating the need for imputation.
It has been shown that generally moderated t-tests perform much better also in proteomics data, as compared to t-tests (Kammers et al. 2015). Therefore, we will use a moderated t-test in this example.
Please note that in this example we are not imputing missing values. You can, however, do so by using the function impute(). This function uses the output of assign_missingness() as its input. You can use two different imputation methods:
• method = ludovic will sample values that are MNAR from a normal distribution around a value that is 3 (log2) lower than the mean intensity of the non-missing condition. The method is was developed by our colleague Ludovic Gillet.
• method = noise will sample MNAR values from a normal distribution around the mean noise of the complete condition. This requires you to have an additional column with information on the noise, which can be obtained from Spectronaut.
Both methods impute MAR data using the mean and variance of the condition with the missing data. Missingness assigned as NA will not be imputed.
Note: If data is imputed this can lead to invalid inferential conclusions due to underestimating statistical uncertainty or it can cause loss of statistical power (Ahlmann-Eltze et al. 2020). Therefore, we do not recommend using a moderated t-test or the proDA algorithm after imputation.
Since we are dealing with a LiP-MS data set we perform the statistical analysis on the precursor* level. For protein abundance data you can simply use protein abundances as your intensities and select your protein groups column for the grouping argument. Make sure to retain any columns you need for further data analysis with the retain_columns argument of both functions.
_*A peptide precursor is the actual molecular unit that was detected on the mass spectrometer. This is a peptide with one specific charge state and its modification(s)._
Note: Although it is not required for the data set analysed in this vignette, analysis of LiP-MS data frequently requires correction of LiP peptide intensities for changes in protein abundance. This can be done using the steps outlined in Schopper et al. 2017.
diff_abundance_data <- data_filtered_uniprot %>%
assign_missingness(
sample = r_file_name,
condition = r_condition,
grouping = eg_precursor_id,
intensity = normalised_intensity_log2,
ref_condition = "control",
completeness_MAR = 0.7,
completeness_MNAR = 0.25,
retain_columns = c(pg_protein_accessions,
go_molecular_function,
database_string,
start,
end,
length,
coverage)
) %>%
calculate_diff_abundance(
sample = r_file_name,
condition = r_condition,
grouping = eg_precursor_id,
intensity_log2 = normalised_intensity_log2,
missingness = missingness,
comparison = comparison,
method = "moderated_t-test",
retain_columns = c(pg_protein_accessions,
go_molecular_function,
database_string,
start,
end,
length,
coverage)
)
### p-value distribution
The p-value calculated with the moderated t-test is automatically adjusted for multiple testing using the Benjamini-Hochberg correction. This assures that we keep the false discovery rate low. An assumption of this correction is however, that p-values should have an overall uniform distribution. If there is an effect in the data, there will be an increased frequency of low p-values. You can check this by using the protti function pval_distribution_plot(). This also helps you assess whether your p-value distribution fulfills the assumptions for your selected FDR control. The cp4p R package is another great way to check the assumptions underlying FDR control in quantitative experiments.
pval_distribution_plot(data = diff_abundance_data,
grouping = eg_precursor_id,
pval = pval
)
For this subset of data the distribution of p-values is relatively flat and there is no large increase in values in the low p-value range (the distribution is uniform when a lot of your hypotheses are null). This is likely because, for this experiment, only a very small fraction of peptides show changes.
It is recommended to always check the (non-adjusted) p-value distribution. A histogram that does not have a uniform distribution with or without a specific enrichment for very low p-values (due to a treatment induced effect) indicates a failure of the theoretical null distribution, which could have several causes (Efron 2010, chapter 6).
### Volcano plot
Next we are going to visualise the output of the previously performed hypothesis test to assess the results of our experiment. For this we are going to plot a volcano plot with fold-changes on the x-axis and the p-value on the y-axis. The output of the previously used calculate_diff_abundance() function is ideal to use for the volcano_plot() function as it contains all the information we need: precursor IDs, protein IDs, fold changes (diff), p-values (pval) and adjusted p-values (adj_pval). We are going to highlight the peptides of the known target of rapamycin FKBP12 (UniProt ID = P62942) in blue to quickly find the peptides in the plot. You can also make the plot interactive by setting interactive = TRUE. This will help you quickly obtain more information on each point in the plot.
Since adjusted p-values are related to unadjusted p-values often in a complex way, it makes them hard to be interpret if they would be used for the y-axis. To nevertheless use the information of adjusted p-values for the plot, you can provide the column name of the adjusted p-values to the significance_cutoff argument next to the desired cutoff. The function will look for the closest adjusted p-values above and below the set cutoff and take the mean of the corresponding p-value as the cutoff line. If there is no adjusted p-value in the data that is below the set cutoff no line is displayed. This allows you to display volcano plots using p-values while using adjusted p-values for the cutoff criteria.
volcano_plot(
data = diff_abundance_data,
grouping = eg_precursor_id,
log2FC = diff,
significance = pval,
method = "target",
target_column = pg_protein_accessions,
target = "P62942",
x_axis_label = "log2(fold change) Rapamycin treated vs. untreated",
)
# The significance_cutoff argument can also just be used for a
# regular cutoff line by just providing the cutoff value, e.g.
# signficiance_cutoff = 0.05
### Barcode plot
For LiP-MS experiments a good way to see where on the protein the changes due to binding or conformational changes occur is to plot a barcode plot. A barcode plot can be created with the protti function barcode_plot(). The detected peptides are coloured in grey and the changing peptides are highlighted in blue.
In order to produce a barcode plot only for our target FKBP12 we create a data frame that contains only information for our target protein using dplyr’s filter() function. The filtered data frame is then used as the input for the plot.
FKBP12 <- diff_abundance_data %>%
filter(pg_protein_accessions == "P62942")
barcode_plot(
data = FKBP12,
start_position = start,
end_position = end,
protein_length = length,
coverage = coverage,
colouring = diff,
cutoffs = c(diff = 1, adj_pval = 0.05),
protein_id = pg_protein_accessions
)
### Wood’s plot
An additional way to plot LiP-MS changes is the Woods’ plot. This plot will show the extent of the precursor fold changes along the protein sequence. The precursors are located on the x-axis based on their start and end positions. The position on the y-axis displays the fold change. The vertical size (y-axis) of the box representing the precurors does not have any meaning.
To produce a Woods’ plot we use the function woods_plot() and colour the peptides according to their adjusted p-values. We are highlighting significant adjusted p-values (< 0.01) with an asterisk. Peptides can also be coloured by another categorical or continous variable. Asterisks can be added for any logical (binary) variable.
FKBP12 <- FKBP12 %>%
mutate(significant = ifelse(adj_pval < 0.01, TRUE, FALSE))
woods_plot(
data = FKBP12,
fold_change = diff,
start_position = start,
end_position = end,
protein_length = length,
coverage = coverage,
protein_id = pg_protein_accessions,
facet = FALSE,
fold_change_cutoff = 1,
highlight = significant
)
### Peptide profile plots
To see how the individual precursors in our target protein are changing with the treatment we plot profile plots by using the function peptide_profile_plot(). This is particularly useful as you can quickly see if your whole protein changes in abundance or only a fraction of precursors/peptides. If you have protein abundance data you can also use the plot to show changes in protein abundance over your treatment condition(s). By selecting multiple targets (as a vector) you can produce the plot for multiple proteins.
FKBP12_intensity <- data_filtered_uniprot %>%
filter(pg_protein_accessions == "P62942")
peptide_profile_plot(
data = FKBP12_intensity,
sample = r_file_name,
peptide = eg_precursor_id,
intensity_log2 = normalised_intensity_log2,
grouping = pg_protein_accessions,
targets = "P62942",
protein_abundance_plot = FALSE
)
protti includes additional helpful functions that do not make sense to use for this data set but apply to data sets of full size that have global changes. These functions include the calculate_go_enrichment()function that helps you check if your hits are enriched for specific gene ontology (GO) terms, the analyse_functional_network() function that plots a String network based on information from StringDB for your hits and the calculate_kegg_enrichment() function which checks for enriched pathways in your hits. Furthermore, you can directly check for enrichment of a self defined treatment with the calculate_treatment_enrichment() function.
For GO enrichment you would add an additional column to your data frame containing information on whether your hit is significant or not. You can do this by using the dplyr function mutate(). Here we want the column to contain logicals that are either TRUE when the adjusted p-value is below 0.05 and the log2(fold change) is below -1 or above 1 or to be FALSE if this is not the case. We use the ifelse() function to produce the logicals. Furthermore, we annotate if the hit is true positive by marking peptides of the known rapamycin binding protein FKBP12.
For the network analysis we filter the previously produced data frame containing the is_significant column for significant hits. This data frame can then be used as an input for analyse_functional_network() to check if the proteins can be found in an interaction network based on StringDB information.
For calculate_kegg_enrichment() you need to first use the function fetch_kegg() to obtain the KEGG pathway identifiers for your data set. You can then use dplyr’s right_join()to join the output with the previously produced data frame containing a column indicating whether your hits are significant or not.
If you know all known interactors of your specific treatment you can check for an enrichment of these with the calculate_treatment_enrichment() function. This is particularly useful if your treatment has an effect on many proteins.
diff_abundance_significant <- diff_abundance_data %>%
# mark significant peptides
mutate(is_significant = ifelse((adj_pval < 0.01 & abs(diff) > 1), TRUE, FALSE)) %>%
# mark true positive hits
mutate(binds_treatment = pg_protein_accessions == "P62942")
### GO enrichment using "molecular function" annotation from UniProt
calculate_go_enrichment(
data = diff_abundance_significant,
protein_id = pg_protein_accessions,
is_significant = is_significant,
go_annotations_uniprot = go_molecular_function
)
### Network analysis
network_input <- diff_abundance_significant %>%
filter(is_significant == TRUE)
analyse_functional_network(data = network_input,
protein_id = pg_protein_accessions,
string_id = database_string,
binds_treatment = binds_treatment,
organism_id = 9606)
### KEGG pathway enrichment
# First you need to load KEGG pathway annotations from the KEGG database
# for your specific organism of interest. In this case HeLa cells were
# used, therefore the organism of interest is homo sapiens (hsa)
kegg <- fetch_kegg(species = "hsa")
# Next we need to annotate our data with KEGG pathway IDs and perform enrichment analysis
diff_abundance_significant %>%
# columns containing proteins IDs are named differently
left_join(kegg, by = c("pg_protein_accessions" = "uniprot_id")) %>%
calculate_kegg_enrichment(protein_id = pg_protein_accessions,
is_significant = is_significant,
pathway_id = pathway_id,
pathway_name = pathway_name)
### Treatment enrichment analysis
calculate_treatment_enrichment(diff_abundance_significant,
protein_id = pg_protein_accessions,
is_significant = is_significant,
binds_treatment = binds_treatment,
treatment_name = "Rapamycin")
|
2022-01-27 11:59:23
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https://www.scienceforums.net/topic/20306-the-great-debate-over-dating-systems/#comment-284318
|
# The Great Debate over Dating Systems
## Recommended Posts
For lack of a better suited forum (and not wishing to couch this discussion in religious terms only) I posted this here instead.
http://studentsfriend.com/feed/topic11.html
For myself, if we are to abandon the traditional designations of Before Christ and Anno Domini, then I would favour the use of the same letters (BC and AD) to mean Before Commong Dating and After Common Dating, perhaps going so far as to use BCD and ACD. Failing that, the use of + and - dates seems overall to be a simple and elegant solution, but perhaps only as a form of shorthand.
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I once tried to make a conceptual dating system, for my own use. It was essentially build around different periods in human history.
Dates from before the first true hominids was treted like BC, in the way that the numbers were negative. I called it BH, Before Humanity. It was originally going to be Before Man but I changed it because there is another, more crude meaning for BM.
Then all of the time afterwards was called HE, for Human Era. The H.E. was then divided into three sub-periods.
CE for civilized era, which started approximately the time of the fist cities,
ME for Machine era, starting the generally agreed on date for the industrial revolution.
According to this we would be living during ME.
I never showed this to anyone because of a few denotative problems with the system. As you probably realized there was probably civilization in some form or another before the first cities were build. Also there were machines before the industrial revolution.
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I think we should just all switch to unix timestamps... Far easier...
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I think we should just all switch to unix timestamps... Far easier...
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As you probably realized there was probably civilization in some form or another before the first cities were build. Also there were machines before the industrial revolution.
Indeed. Interesting idea though, to dilineate time by human development.
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I think BCE and CE work fine. My history classes have used these.
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heck why bother beating around the bush and just call it BC and AD, we all know that thats what the dates are really supposed to mean anyway
urrgggghhhhh
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Indeed. Interesting idea though, to dilineate time by human development.
In a sense that's what BC and AD do now, if only for the fact that using that particular dating system is a major development for the past few thousand years. Besides, dividing our history into too many segments would just be confusing. And we always have problems with what constitutes development. Does the western world have to be changing to consitute a new time period or the eastern world? Cause they didn't always go through developmental periods together.
heck why bother beating around the bush and just call it BC and AD, we all know that thats what the dates are really supposed to mean anyway
Agreed.
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Unix time (properly, POSIX time) is the numerical representation of time invented for use with the Unix operating system but used with many other systems. If we stopped bothering with this whole buisness of twisting our lives around the arbitary frames of day and night then Klaynos might have a point.
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twisting our lives around the arbitary frames of day and night then Klaynos might have a point.
Any time system used by humans in general should be based around night and day. They are the principle basis of our cycadian rythm, and day to day lifestyles.
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But night and day compose our basic perseptions of reality.
Doesn't mine, how can something so inconsistent form one of the axonims for your worldview? Go onto like, any IRC channel and try to form an agreement as to when day and night is, see? It just doesn't work!
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Doesn't mine, how can something so inconsistent form one of the axonims for your worldview? Go onto like, any IRC channel and try to form an agreement as to when day and night is, see? It just doesn't work!
Worldveiw?
How is that relivant? At any given moment I only need to know my local time, whats going on in China is irrelivant.
It is generaly agreed that humans tend to function better during the day because of partly overcomable cyrcadian rythms (nightshifts) and because the availability of light is greater. So our perception of time is based on when we are active (usualy during the day) and when we sleep (usualy at night)
It isn't that complicated to adjust your watch while traviling, or to try to call someone in another country at a time appropriate for them, so why would we ever need to use one planet wide constant for time?
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Unix time (properly, POSIX time) is the numerical representation of time invented for use with the Unix operating system but used with many other systems. If we stopped bothering with this whole buisness of twisting our lives around the arbitary frames of day and night then Klaynos might have a point.
Just to add to that, time 0 is jan 1st at 00:00:00 in 1970 GMT
http://en.wikipedia.org/wiki/POSIX_time for more....
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It isn't that complicated to adjust your watch while traviling, or to try to call someone in another country at a time appropriate for them, so why would we ever need to use one planet wide constant for time?
It's not that complicated, but hey, we're human. We revel in spending all types of resources on things that make our lives so easy that it can practically kill us.
Since the invention of the electric lightbulb and of those-things-that-people-where-over-thier-eyes-on-long-distance-flights it has been plenty possible for us to chose exactly what light is avaliable. Shifting our body clocks to fit this would be no worse than ordinary jet-lag, and we'd only ever have to do it once.
It would also mean that we wouldn't be pissing about with daylight saving times and all that rubbish.
Also remember the awesome chain (like the food chain, but awesome).
$Awesomeness \to Unix \to POSIX time$
I don't think I need any further arguments.
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Since the invention of the electric lightbulb and of those-things-that-people-where-over-thier-eyes-on-long-distance-flights it has been plenty possible for us to chose exactly what light is avaliable.
So your suggesting that all human perceptions of time be dependant on a system that works electronically and that we can function by artifically controlling our light.
In my mind this means you are for a heavy dependance on modern technology. Where as my ideal is that civilisation at large uses technology, but is capable of functioning without it.
Personally I am against that for various reasnons. For one, I want to be as different from the creature in my avatar as possible.
Also most scientific systems of measurement are built around natural phenomino, such as milliter of water is = one gram of water. And C tepeature readings go from 0 = Freezing water to 100 = boiling water. It only makes sense to use natural cycles for timing,
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So your suggesting that all human perceptions of time be dependant on a system that works electronically and that we can function by artifically controlling our light.
Well I wouldn't mind' date=' I was merely making a suggestion for those that have something against candlelight and/or just making do with the dark, there are plenty of things that can still get done in the dark y'know.
For one, I want to be as different from the creature in my avatar as possible.
To quote the offending speciesv themselves, "Daleks have no concept of elegance!" so surely they would be the ones using the duodecimal system rather than the beautiful simplicity of POSIX time.
Also most scientific systems of measurement are built around natural phenomino...
Naturalistic fallacy.
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To quote the offending speciesv themselves' date=' "Daleks have no concept of elegance!" so surely they would be the ones using the duodecimal system rather than the beautiful simplicity of POSIX time.
[/quote']
I was refurring to the Dalek dependance on technology, such as computers. A timekeeping system based on an electronic computer is subject to all the limitations of a clock and a computer. So it would be renderd ineffective by black out, EMP, computer viruses, and hackers. It would also follow the typical pattern of a clock which is to gain or lose seconds over time.
Naturalistic fallacy.
Definition of fallacy:
A fallacy is a component of an argument that is demonstrably flawed in its logic or form,
I see no flaw with my logic. My assertion is that basing a timing system for a human lifestyle on a daily phenominon makes sense because Humans are animals that base there lifstyle of their cycadian rythm. This rythm is a daily cycle that is influenced by night/day, we are not machines, nor do I think we should strive to become machines
A day night cycle that we would otherwise follow naturaly is likely to be most compatable with our biology, and psycology.
I'm sure that your timing system can easily be used when working in conjunction with computers. However large portions of the earths population are computer illiterate. Such as people living in third world countries and my grandmother. The old system would prove by far the superior to them whose live don't revolve around machines.
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I was refurring to the Dalek dependance on technology, such as computers. A timekeeping system based on an electronic computer is subject to all the limitations of a clock and[/b'] a computer.
No it's not, just of the timekeeper.
So it would be renderd ineffective by black out, EMP, computer viruses, and hackers.
One of my clocks (the one that makes the tea and switches on the radio) already is electronic, thus computerised but niether does it nor is capable of any network access so I severly doubt that unless a whatever-the-mains-supply-equilivant-to-a-phreaker-is decides to attack my home, the first two would be an issue and as for EMP well, o.k. most of my clocks would be pretty screwed in that event but that wouldn't have anything to do with the time system they were employing.
It would also follow the typical pattern of a clock which is to gain or lose seconds over time.
Of course, but why is this an issue?
I see no flaw with my logic...
The naturallistic fallacy (seriously, look up the whole term, not just the individual words) is the assertion that the way things are is the way things should be, drawing parallels with other systems is commiting said fallacy.
we are not machines, nor do I think we should strive to become machines
Machines merely do what there owners ask of them, the human mind is capable of a lot more. Sticking to some arbitarly designed system, just-because, is about as machine like as you can get.
A day night cycle that we would otherwise follow naturaly is likely to be most compatable with our biology, and psycology.
That's no reason why we shouldn't use the same cycle in a nature that would be compatable with our culture as well.
I'm sure that your timing system can easily be used when working in conjunction with computers.
It does get used with computers, lots of computers.
However large portions of the earths population are computer illiterate. Such as people living in third world countries and my grandmother.
You hardly need a computer to manage a number, if you did then by the same logic I could say that all mathematical ideas devised with the assistance of an abacus is not worth using because so few people now can use abaci.
The old system would prove by far the superior to those[/u'] whose lives don't revolve around machines.
But why? It's more complicated. It makes it harder to work out simple differences (at least for a child who hasn't had much practice). And it's own units don't even fit into each other!
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You have been using yourself alot as an example, so I'll use myself
I would never use the Unix system in my daily life for one reason. It is unfamiliar.
I don't think in terms of unix, ever since I was a child it has always been 5 o'clock, 4 thirty, two hours, eight minutes.
The current system is already well established, people understand it, trying to change it in daily life would be extreamly difficult because it is alien to massive numbers of people who don't work directly with computers.
Your system has some advantages, but I don't program computers, I don't telecomunicate with people from asia, and I don't find it difficult to change my clocks by an hour every once and a while.
So I guess I'm conceeding to you, your system is superior in many ways,
but how would Big Ben look with a Unix face instead of a regular one?
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People worried about decimalising the British currency and switching to metric mesurements, but they got over it in pretty much one generation. Sure it's unfamiliar but why should that be a problem? We live in a world were people can learn new stuff, so why shouldn't we?
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People worried about decimalising the British currency and switching to metric mesurements, but they got over it in pretty much one generation. Sure it's unfamiliar but why should that be a problem? We live in a world were people can learn new stuff, so why shouldn't we?
I live in the USA.
Recall that when we tried to switch to the metric system it didn't work out so well.
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Hey, I always thought that time should be decimalised, I just can’t work in hours, minutes and seconds, I just never got the hang of it.
I can’t see why we should not have 10 ‘Dhours’ in a day, 100 ‘Dminutes’ in a ‘Dhour’ 100 ‘Dseconds’ in a ‘Dminute’.
This decimalisation could be extended to the rest of the calendar - 5 days in a ‘demiDweek’ and 10 ‘Dmonths’ in a year. There could be 35 days (7 ‘dDweeks’) in each the 10 Dmonths with an extra 3 ‘Dweeks’ (15days) making up a special part month for the ‘Dholidays’ at the end of the year.
Also, the last day of the year should be the longest day in the N hemisphere.
And noon on the longest day calculated at the meridian at which the Sun is directly above at the time of the Earths axis being tilted to its maximum on that day. Synchronise your watches everyone
OK so there are still only approximately 365 days in a year and corrections would need to be made to keep in sync with the average of the earths spin and orbit, but it’s that simple really
As for the BC AD system – Pick a year, any one will do (I would prefer a specific celestial alignment to signify this, as most of the rest of our date and time system is based on this type of thing), call that year 0 (not 1, Duh) and start from there, before that is – after that is +.
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I live in the USA.
Recall that when we tried to switch to the metric system it didn't work out so well.
That's because we are too stubborn and lazy to adopt a better system.
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That's because we are too stubborn and lazy to adopt a better system.
GOD bless America!
From what I've heard it was more a matter of conversions. People never just used the metric measurments. They were always trying to convert the metric to the standard.
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2022-06-30 16:52:51
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https://www.gradesaver.com/textbooks/math/algebra/college-algebra-6th-edition/chapter-1-equations-and-inequalities-exercise-set-1-7-page-198/122
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College Algebra (6th Edition)
Published by Pearson
Chapter 1 - Equations and Inequalities - Exercise Set 1.7 - Page 198: 122
Answer
One must drive $more$ $than$ $100$ miles for Basic to be a better deal.
Work Step by Step
First, we model the cost of each rental with the following equations: $$Basic_{cost} = 50 + 0.2m$$ $$Continental_{cost} = 20 + 0.5m$$ where $m$ represents the amount of miles to be driven. The exercise asks for the amount of miles needed for Basic to be a better deal than Continental. In other words, at what $m$ Continental becomes greater than Basic: $$Continental_{cost}\gt Basic_{cost}$$ $$20 + 0.5m \gt 50 + 0.2m$$ By solving for $m$: $$0.5m - 0.2m \gt50 - 20$$ $$0.3m \gt 30$$ $$m\gt 100$$
After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.
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2019-02-20 03:19:16
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https://socratic.org/questions/how-do-you-solve-x-80-19
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How do you solve x + 80 = -19?
Jan 20, 2016
$x = - 99$
Explanation:
$x + 80 = - 19$
move from LHS to the RHS all the therms without the $x$, changing their sign
$x = - 19 \textcolor{g r e e n}{- 80}$
$x = - 99$
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2019-09-18 18:09:52
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https://en.wikipedia.org/wiki/Riemann_Xi_function
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# Riemann Xi function
Riemann xi function ${\displaystyle \xi (s)}$ in the complex plane. The color of a point ${\displaystyle s}$ encodes the value of the function. Darker colors denote values closer to zero and hue encodes the value's argument.
In mathematics, the Riemann Xi function is a variant of the Riemann zeta function, and is defined so as to have a particularly simple functional equation. The function is named in honour of Bernhard Riemann.
## Definition
Riemann's original lower-case "xi"-function, ${\displaystyle \xi }$ was renamed with an upper-case ${\displaystyle ~\Xi ~}$ (Greek letter "Xi") by Edmund Landau. Landau's lower-case ${\displaystyle ~\xi ~}$ ("xi") is defined as[1]
${\displaystyle \xi (s)={\frac {1}{2}}s(s-1)\pi ^{-s/2}\Gamma \left({\frac {s}{2}}\right)\zeta (s)}$
for ${\displaystyle s\in \mathbb {C} }$. Here ${\displaystyle \zeta (s)}$ denotes the Riemann zeta function and ${\displaystyle \Gamma (s)}$ is the Gamma function.
The functional equation (or reflection formula) for Landau's ${\displaystyle ~\xi ~}$ is
${\displaystyle \xi (1-s)=\xi (s)~.}$
Riemann's original function, rebaptised upper-case ${\displaystyle ~\Xi ~}$ by Landau,[1] satisfies
${\displaystyle \Xi (z)=\xi \left({\tfrac {1}{2}}+zi\right)}$,
and obeys the functional equation
${\displaystyle \Xi (-z)=\Xi (z)~.}$
Both functions are entire and purely real for real arguments.
## Values
The general form for positive even integers is
${\displaystyle \xi (2n)=(-1)^{n+1}{\frac {n!}{(2n)!}}B_{2n}2^{2n-1}\pi ^{n}(2n-1)}$
where Bn denotes the n-th Bernoulli number. For example:
${\displaystyle \xi (2)={\frac {\pi }{6}}}$
## Series representations
The ${\displaystyle \xi }$ function has the series expansion
${\displaystyle {\frac {d}{dz}}\ln \xi \left({\frac {-z}{1-z}}\right)=\sum _{n=0}^{\infty }\lambda _{n+1}z^{n},}$
where
${\displaystyle \lambda _{n}={\frac {1}{(n-1)!}}\left.{\frac {d^{n}}{ds^{n}}}\left[s^{n-1}\log \xi (s)\right]\right|_{s=1}=\sum _{\rho }\left[1-\left(1-{\frac {1}{\rho }}\right)^{n}\right],}$
where the sum extends over ρ, the non-trivial zeros of the zeta function, in order of ${\displaystyle |\Im (\rho )|}$.
This expansion plays a particularly important role in Li's criterion, which states that the Riemann hypothesis is equivalent to having λn > 0 for all positive n.
A simple infinite product expansion is
${\displaystyle \xi (s)={\frac {1}{2}}\prod _{\rho }\left(1-{\frac {s}{\rho }}\right),\!}$
where ρ ranges over the roots of ξ.
To ensure convergence in the expansion, the product should be taken over "matching pairs" of zeroes, i.e., the factors for a pair of zeroes of the form ρ and 1−ρ should be grouped together.
## References
1. ^ a b Landau, Edmund (1974) [1909]. Handbuch der Lehre von der Verteilung der Primzahlen [Handbook of the Study of Distribution of the Prime Numbers] (Third ed.). New York: Chelsea. §70-71 and page 894.
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2022-01-27 09:39:21
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https://physics.stackexchange.com/questions/314384/what-defines-a-large-gauge-transformation-really
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# What defines a large gauge transformation, really?
Usually, one defines large gauge transformations as those elements of $SU(2)$ that can't be smoothly transformed to the identity transformation. The group $SU(2)$ is simply connected and thus I'm wondering why there are transformations that are not connected to the identity. (Another way to frame this, is to say that large gauge transformations can not be built from infinitesimal ones.)
An explicit example of a large gauge transformation is
$$$$U^{\left( 1\right) }\left( \vec{x}\right) =\exp\left( \frac{i\pi x^{a}\tau^{a}}{\sqrt{x^{2}+c^{2}}}\right)$$$$
How can I see explicitly that it is impossible to transform this transformation to the identity transformation?
I can define
$$U^\lambda(\vec x) = \exp\left( \lambda \frac{i\pi x^{a}\tau^{a}}{\sqrt{x^{2}+c^{2}}}\right)$$
and certainly
$$U^{\lambda=0}(\vec x) = I$$ $$U^{\lambda=1}(\vec x) = U^{\left( 1\right) }\left( \vec{x}\right)$$
Thus I have found a smooth map $S^3 \to SU(2)$ that transforms $U^{\left( 1\right) }\left( \vec{x}\right)$ into the identity transformation. So, in what sense is it not connected to identity transformation?
Framed differently: in what sense is it true that $U^{\lambda=1}(\vec x)$ and $U^{\lambda=0}(\vec x)$ aren't homotopic, although the map $U^\lambda(\vec x)$ exists? My guess is that at as we vary $\lambda$ from $0$ to $1$, we somehow leave the target space $SU(2)$, but I'm not sure how I can see this.
In addition, if we can write the large gauge transformation as an exponential, doesn't this does mean explicitly that we get a finite large gauge transformation, from infinitesimal ones?
According to this paper, the defining feature of large gauge transformations is that the function in the exponent $\omega(x)$ is singular at some point. Is this singularity the reason that we can't transform large gauge transformations "everywhere" to the identity transformations? And if yes, how can we see this?
Edit: I got another idea from this paper. There, the authors state that its not enough that we find a map $U^\lambda(\vec x)$, with the properties mentioned above, but additionally this map must have the following limit $$U^\lambda(\vec x) \to I \quad \text{ for } x\to \infty \quad \forall \lambda.$$ Obviously, this is not correct for my map $U^\lambda(\vec x)$. However, I don't understand why we have here this extra condition.
Edit 2: As mentioned above, there only exists no smooth map between $U^{\lambda=1}(\vec x)$ and $U^{\lambda=0}(\vec x)$, if we restrict ourselves to those gauge transformations that satisfy
$$U(x) \to I \quad \text{ for } x\to \infty.$$
The mystery therefore is, why we do this. It seems, I'm not the only one puzzled by this, because Itzykson and Zuber write in their QFT book:
"there is actually no very convincing argument to justify this restriction".
• For the exponential, you need to start thinking about bundles and simultaneous local descriptions, as in the answer below. For the "continuously connected to identity" part of the question, you can think in terms of physical states: "small" gauge transformations leave you in the vacuum, "large" ones take you over sphaleron barriers to different vacua. – Demosthene Feb 24 '17 at 17:25
# Bundles and compactified spacetime
A gauge theory cannot be looked at purely locally, it has inherently global features on cannot see locally. The proper mathematical formalization of a Yang-Mills gauge theory is that the gauge field $A$ is a connection on a principal bundle $P\to M$ over spacetime $M$. However, in practice, it turns out that physicists don't actually want $M$ to be spacetime itself, but spacetime compactified.
We can see this most clearly in the construction of the BPST instanton on Euclidean $\mathbb{R}^4$: The gauge invariant trace of the field strength itself goes as $\mathrm{tr}(F)\propto \frac{\rho^2}{(x^2+\rho^2)^2}$ and is well-defined everywhere, falling off toward infinity. But if we consider the associated gauge potential $A$, one finds it is not well-defined everywhere, it goes as $A\propto \frac{x^3}{x^2(x^2+\rho^2)}$, which is singular for $x\to 0$, yet well-defined for $\lvert x\rvert\to\infty$ as $A(x)\to U(x)^{-1}\mathrm{d}U(x)$, where $U(x)$ is essentially the gauge transformation you wrote down in your question.
So we want $F$ as a physically allowed field strength, yet its corresponding $A$ is not well-defined on $\mathbb{R}^4$. The bundle viewpoint cannot help us because all bundles over Euclidean space are trivial, meaning $A$ must always be defined globally. However, if we pass to $S^4$ as the conformal compactification of $\mathbb{R}^4$ and identify one of the poles with "infinity" and the other with zero, then non-trivial bundles because possible, and we get two local descriptions on the northern and southern "hemispheres" which we can usually extend over the entire sphere except a single point. If the local description of $A$ extends over the entire sphere, then the principal bundle of the gauge theory is trivial.
But we already saw that the specific $A$ we chose does not extend to $x=0$, and in fact the topological invariant $\int\mathrm{tr}(F\wedge F)$ is non-zero, meaning the bundle is non-trivial, meaning $A$ cannot extend over the entire sphere. In particular, it is inherently impossible to find a $A$ that is well-defined at every $x\in\mathbb{R}^4$ and has a well-defined limit towards infinity that gives us the BPST instanton solution $F$.
So you have exactly two choices: Either we must consider gauge theory on $S^4$ instead on $\mathbb{R}^4$, or the BPST instantons - all instantons, in fact - are not actually allowed solutions of the gauge theory. Standard physics opts for the former, in light of instanton contributions to detectable things like the axial anomaly.
# Large gauge transformations
Now that we know that we are looking at a principal bundle $P\to S^4$, a gauge transformation is a fiber-preserving automorphism $P\to P$, and it can happen that these are not homotopic to the identity map $P\to P$. As a toy example, consider the $\mathrm{U}(1)$-bundle $\mathrm{U}(1)\times S^1\to S^1$, which is the torus, and the gauge transformation $\mathrm{U}(1)\times S^1\to\mathrm{U}(1)\times S^1, (g,s)\mapsto (gs,s)$, which takes hte canonical embedding $S^1\to \mathrm{U}(1)\times S^1$ and winds it once around the circle $\mathrm{U}(1)$. Since the winding number is a homotopy invariant, the image of the $S^1$, as a path, is not homotopic to the source and therefore this transformation is not homotopic to the identity. This is a large gauge transformation in the proper, mathematical sense, as defined in the Wikipedia article and discussed, for instance, in this answer by David Bar Moshe. I am actually not certain whether there are "true" large gauge transformations on $S^4$ in this sense, but I believe there aren't.
# "False gauge transformations", or: Transition functions
By lacking the formal machinery of principal bundles, the physicist often confuses two different objects - the gauge transformations $P\to P$, which descend to functions $g_i : U_i\to G$ in the local description, and the transition function, which are gauge transformation-like functions $t_{ij} : U_i\cap U_j\to G$ that define the bundle in the local description and do not exist globally. Both the $g_i$ and the $t_{ij}$ fulfill certain compatibility conditions to be globally well-defined.
Now, if the physicist makes a gauge transformation, they usually only consider $\mathbb{R}^4$, meaning they implicitly set the gauge transformation on the other local description - the open set around $\infty$ - to be trivial. The compatibility condition then says that $g_i = t_{ij}$ on $U_i\cap U_j$. In the physicist's local description, this overlap is the sphere at infinity, i.e the behaviour of the gauge transformation as $\lvert x\rvert \to \infty$. So the condition that $U(x)\to I$ that confuses you, Itzykon, Zuber and probably countless others is nothing but the condition that the $U(x)$, given in this local description, actually lifts to a proper gauge transformation on the bundle $P\to S^4$.
A $U(x)$ that does not do this either needs to be complimented by its corresponding transformation in the other local description, or it changes the bundle, that is, the physicist has declared that it changed the transition function, and thereby (probably) the bundle. The $\mathrm{SU}(2)\cong S^3$ bundles over $S^4$ are classified by maps $S^3\to S^3$ "on the equator", in parfect analogy to $\mathrm{U}(1)$-bundles on $S^1$ as described in this answer of mine. And, as $x\gg c$, your $U^{(1)}(x)$ becomes a function $$\frac{x}{\lvert x\rvert}\mapsto \exp\left(\frac{\mathrm{i}\tau_\mu x^\mu}{\lvert x\rvert}\right),$$ where $x/\lvert x\rvert$ is just a point on the unit sphere $S^3\subset\mathbb{R}^4$ and the r.h.s. lies in $\mathrm{SU}(2)\cong S^3$ naturally. So this is a map $S^3\to S^3$ whose homotopy class classifies the bundle, and it is not too hard to see that it winds the $S^3$ once around itself, in contrast to the constant map, so the implicit change it carries out on the transition function actually changes the bundle. It is not a gauge transformation for the theory on $S^4$ since it does so, not even a "large" one, but these transformations are also often called large gauge transformations. Note finally that it is also not an allowed gauge transformation on $\mathbb{R}^4$ since it is not smooth at $0$.
• – JakobH Mar 16 '17 at 11:17
• +1 Glad that this is finally being picked up and propagated! – Urs Schreiber Dec 9 '17 at 17:49
• Do you mean that $\tau_\mu x^\mu = x^0 \mathbb{I} + \sum_{i=1}^3 x^i \tau_i$? Because in that case your expression for the limit of $U^{(1)}$ is not an element of $SU(2)$ because of its determinant. I think the expression $x^a \tau^a$ in the question is supposed to mean just $\sum_{i=1}^3 x^i \tau_i$, which is an element of $SU(2)$. But then the limit of $U^{(1)}$ is a map from $S^2$. – Friedrich Dec 21 '17 at 18:15
• @Friedrich I think the expression should just be $x^a \tau^a$ as in the question, but, no, it is not a map from $S ^2$ - note that three of the four components of a vector in $S^3$ already almost uniquely identify the point - the constraint of being a unit vector means that $x^0$ is determined up to a sign, so this is actually a map from a hemisphere of $S ^3$ to $S ^3$ (And extendible to a map $S ^3 \to S ^3$). However it's not as easy as I claim to see its winding number now, I think. – ACuriousMind Dec 22 '17 at 12:41
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2019-05-24 16:01:01
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http://fluidsengineering.asmedigitalcollection.asme.org/article.aspx?articleid=2717099
|
0
Research Papers: Flows in Complex Systems
# Experimental Investigation of the Swirl Development at the Inlet of a Coaxial Rotating Diffuser or Nozzle
[+] Author and Article Information
Ferdinand-J. Cloos
Chair of Fluid Systems,
Department of Mechanical Engineering,
Otto-Berndt-Str. 2,
Peter F. Pelz
Professor
Chair of Fluid Systems,
Department of Mechanical Engineering,
Otto-Berndt-Str. 2,
1Corresponding author.
Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received February 6, 2018; final manuscript received November 12, 2018; published online January 29, 2019. Assoc. Editor: Olivier Coutier-Delgosha.
J. Fluids Eng 141(4), 041107 (Jan 29, 2019) (8 pages) Paper No: FE-18-1084; doi: 10.1115/1.4042095 History: Received February 06, 2018; Revised November 12, 2018
## Abstract
When a fluid enters a rotating pipe, a swirl boundary layer with thickness of $δ̃S$ appears at the wall and interacts with the axial momentum boundary layer with thickness of $δ̃$. The swirl is produced by the wall shear stress and not due to kinematic reasons as by a turbomachine. In the center of the pipe, the fluid is swirl-free and is accelerated due to axial boundary layer growth. Below a critical flow number φ < φc, there is flow separation, known in the turbomachinery context as part load recirculation. The previous work analyzes the flow at the inlet of a coaxial rotating circular pipe ($R̃=R̃0$). For a systematic approach to a turbomachine, the influence of the turbine's and pump's function, schematically fulfilled by a diffuser and a nozzle, on the evolution of the swirl and flow separation is to analyze. The radius of the rotating pipe depends linearly on the axial coordinate, yielding a rotating circular diffuser or nozzle. The swirl evolution depends on the Reynolds number, flow number, axial coordinate, and apex angle. The influence of the latter is the paper's main task. The circumferential velocity component is measured applying one-dimensional laser Doppler anemometry (LDA) to investigate the swirl evolution.
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## References
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Cloos, F.-J. , Zimmermann, A.-L. , and Pelz, P. , 2016, “ Two Turbulent Flow Regimes at the Inlet of a Rotating Pipe,” Eur. J. Mech. B, 61, pp. 330–335.
Cloos, F.-J. , Zimmermann, A.-L. , and Pelz, P. , 2016, “ A Second Turbulent Regime When a Fully Developed Axial Turbulent Flow Enters a Rotating Pipe,” ASME Paper No. GT2016-57499.
Stapp, D. , 2015, Experimentelle und Analytische Untersuchung zur Drallgrenzschicht (Forschungsberichte zur Fluidsystemtechnik), Technische Universität Darmstadt, Darmstadt, Germany.
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Cloos, F.-J. , and Pelz, P. , 2018, “ Developing Swirl Boundary Layer and Flow Separation and the Inlet of a Coaxial Rotating Diffuser or Nozzle,” ASME J. Fluids Eng., 141(7), p. 071102.
Nishibori, K. , Kikuyama, K. , and Murakami, M. , 1987, “ Laminarization of Turbulent Flow in the Inlet Region of an Axially Rotating Pipe,” Bull. JSME, 30, pp. 255–262.
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Crane, C. , and Burley, D. , 1976, “ Numerical Studies of Laminar Flow in Ducts and Pipes,” J. Comput. Appl. Math., 2(2), pp. 95–111.
Jungnitz, G. , 1949, “ Rechnerische Untersuchung Von Diffusoren,” Forsch. Ingenieurwes., 16(2), pp. 60–62.
Szablewski, W. , 1952, “ Turbulente Strömungen in Konvergenten Kanälen,” Arch. Appl. Mech., 20(1), pp. 37–45.
Szablewski, W. , 1954, “ Turbulente Strömungen in Divergenten Kanälen,” Arch. Appl. Mech., 22(4), pp. 268–281.
Schlichting, H. , and Gersten, K. , 1961, “ Berechnung Der Strömung in Rotationssymmetrischen Diffusoren Mit Hilfe Der Grenzschichttheorie,” Z. Flugwiss., 9(4/5), pp. 136–140.
Börger, G.-G. , 1973, “ Optimierung von Windkanaldüsen für den Unterschallbereich,” Ph.D. thesis, Ruhr-Universität Bochum, Bochum, Germany.
Gersten, K. , and Herwig, H. , 1992, Strömungsmechanik: Grundlagen Der Impuls-, Wärme- und Stoffübertragung Aus Asymptotischer Sicht, Vieweg, Braunschweig/Wiesbaden, Germany.
Seifert, F. , 2006, “ Berechnung Inkompressibler Reibungsbehafteter Ringdiffusorströmungen Nach der Schlankkanaltheorie,” Ph.D. thesis, Ruhr-Universität Bochum, Bochum, Germany.
Taylor, G. , 1950, “ The Boundary Layer in the Converging Nozzle of a Swirl Atomizer,” Q. J. Mech. Appl. Math., 3(2), pp. 129–139.
Binnie, A. , and Harris, D. , 1950, “ The Application of Boundary-Layer Theory to Swirling Liquid Flow Through a Nozzle,” Q. J. Mech. Appl. Math., 3(1), pp. 89–106.
Broujerdi, A. , and Kerbriaee, A. , 2010, “ Pressure Loss of Turbulent Swirling Flow in Convergent Nozzle,” 18th Annual International Conference on Mechanical Engineering (ISME), Tehran, Iran, May 11–13, Paper No. ISME2100-3101.
## Figures
Fig. 3
Laser beam path into the rotating pipe of the 1D laser Doppler anemometry
Fig. 2
Experimental setup with (a) configuration I for a thin axial boundary layer and (b) configuration II for a fully developed axial boundary layer
Fig. 1
Flow through the inlet of a rotating circular diffuser
Fig. 4
Interpolated isolines of uϕ show the development of the swirl boundary layer for configuration I. Measurement points are illustrated by the markers.
Fig. 5
Swirl boundary layer thickness δS02 with various apex angles versus (a) axial coordinate, (b) Reynolds number, and (c) flow number for configuration I
Fig. 6
Swirl boundary layer thickness δS07 with various apex angles versus (a) axial coordinate, (b) Reynolds number, and (c) flow number for configuration II
Fig. 7
Swirl velocity profile for various axial coordinates, Reynolds number, flow number, and apex angle for (a)–(d) configuration I, a thin axial laminar boundary layer, and for (e)–(h) configuration II, a fully developed axial turbulent boundary layer
Fig. 8
Turbulence intensity of the swirl velocity for various axial coordinates, Reynolds number, flow number, and apex angle for (a)–(d) configuration I, a thin axial laminar boundary layer, and for (e)–(h) configuration II, a fully developed axial turbulent boundary layer
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2019-08-19 15:09:17
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https://www.gamedev.net/forums/topic/3917-which-compilers-do-all-of-you-use-/
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#### Archived
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# Which compilers do all of you use ???
This topic is 6783 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.
## Recommended Posts
well which language i use DJGPP for c its free MASM for assembly its free too thats all i know or am learning to program in
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TITAN
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I use Microsoft Visual C++ 6.0 Standard Edition. It has the best IDE i've seen in a compiler, and it works with just about everything. Its not free though, but you can get it pretty cheap (I found it for $50, although usually its about$100). The Proffesional Edition optimizes your code, but it costs a lot more ($200-$400).
One warning: Visual C++ only makes windows code, you can't make any DOS programs with it.
Also, the whole "visual" part is just a name, so don't let that confuse you -- Its just a C++ compiler.
--TheGoop
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If you're going to write Windows programs (I do, especially Windows games, and can afford MSVC++ (Microsoft Visual C++), you should *definitely* get it.
DirectX supports VC++ fully, and I think it comes with Borland compatible libs too, but I see a lot of gossip going around that getting DX to work on Borland takes a little messing around.
Also, Professional version is good to get since there are optimizations that got my last library a good 5 frame rates up. It however doesn't *really* make a difference, but again, that 5 fps was just what I needed to say "its good!"
Rock
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2018-04-26 08:06:41
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https://mathspace.co/textbooks/syllabuses/Syllabus-410/topics/Topic-7301/subtopics/Subtopic-97417/?activeTab=interactive
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NZ Level 7 (NZC) Level 2 (NCEA)
z-scores (what they are and calculating them)
## Interactive practice questions
The mean of a set of scores is $77$77 and
the standard deviation is $29$29. Find the value of:
a
$\text{Mean }-\text{Standard Deviation}$Mean Standard Deviation
b
$\text{Mean }+2\times\text{Standard Deviation}$Mean +2×Standard Deviation
c
$\text{Mean }-\frac{\left(2\times\text{Standard Deviation}\right)}{3}$Mean (2×Standard Deviation)3
d
$\text{Mean }+\frac{\left(4\times\text{Standard Deviation}\right)}{5}$Mean +(4×Standard Deviation)5
e
$\text{Mean }+3\times\text{Standard Deviation}$Mean +3×Standard Deviation
f
$\text{Mean }-2\times\text{Standard Deviation}$Mean 2×Standard Deviation
Easy
Approx 4 minutes
The mean of a set of scores, denoted by $\mu$μ, is $51$51 and
the standard deviation is $16$16, and is denoted by $\sigma$σ. Find the value of:
The literacy rate of a population is used to help measure the level of development of a country. The average literacy rate in a particular country is $59%$59%, and the standard deviation is $5%$5%. The literacy rate varies from place to place within the country.
The following table shows the marks obtained by a student in two subjects.
### Outcomes
#### S7-4
S7-4 Investigate situations that involve elements of chance: A comparing theoretical continuous distributions, such as the normal distribution, with experimental distributions B calculating probabilities, using such tools as two-way tables, tree diagrams, simulations, and technology.
#### 91267
Apply probability methods in solving problems
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2021-09-20 18:06:11
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